Escobar grew up in Cuba until he emigrated to the United States in 2004, washing up on the shores of Miami. Brayan Pena, a former teammate of Escobar with the Atlanta Braves, was Escobar's best friend growing up in Cuba, with the two boys often playing on the same team. Pena defected from Cuba several years before Escobar did. Because of Escobar's close friendship with Pena, the defection of Pena caused many problems for Escobar with Cuban authorities. Escobar was no longer allowed to travel to international tournaments, was constantly interrogated and followed by security guards dispatched by Fidel Castro. Sports Illustrated reported Escobar as saying "I couldn't even go to the bathroom by myself." Frustrated, Yunel would turn on his guard in the street and yell, "NO. ME. VOY. No me sigas más!" (I'm. Not. Going. Stop following me!)
According to Sports Illustrated's accounts of Escobar's own admissions, Escobar was suspended from playing baseball on numerous occasions by Cuban officials.  It was those suspensions, according to Escobar, which finally convinced him to flee Cuba. According to SI, Escobar admitted that he once, out of frustration, threw a ball at a fan which led to one suspension and that he'd been suspended on another occasion for not wearing the right pair of black pants.  Escobar also told SI that he'd been benched on numerous occasions by coaches who questioned his loyalty to Fidel Castro. 
With regard to Escobar's two days at sea while fleeing Cuba, Escobar told SI that he barely spoke, not even to his five teammates from the Havana club Industriales, Cuba's version of the Yankees, for fear of agitating other fellow passengers who were beset by dehydration, hunger and sickness. "Someone might just throw you off the boat if they didn't like you," Escobar said. 
"I was very scared," he recalled. "Just imagine two days at sea, not knowing exactly where you are going. I knew I was going to the United States. Outside of that, I didn't really know.... And I couldn't tell my parents where I was going until I got here."
The rest of Escobar's family would not make it out of Cuba until 2007. His father and sister made it to America in the spring of 2007. His mother made it to the United States several months later in the summer of 2007. Before his mother made it safely to the U.S., Escobar constantly worried about her, fearing that the Cuban government would hold it against her that he left the country without governmental permission. "I'm always thinking about her," he said through a translator to reporters in May 2007. "I will feel much better once she's made it here safely. I'm hoping she'll be the spark I need to take my game to the next level." Back in Havana, his mother was scared as well. "I was very sad by myself there. I was a little afraid. The most beautiful things in my life are Yunel and Yunay. I thought I might never see them again." When neighbors with Internet access informed her that Escobar had been called up to the major leagues, Escobar's mother said, "I was crying and screaming because that was one of his dreams. He risked his life to get it and he finally achieved it."
After obtaining the permission of the Cuban government to leave and go to the U.S. to seek political asylum and finally getting an opportunity to watch her son play at Turner Field in Atlanta, his mother said, "This is the most beautiful thing that can happen to you," she said in Spanish to her son's wife, Minerva Escobar, who translated. "I still can't believe I am talking to him again [in person]. He's not the same boy who left Cuba three years ago." "I thought he was going to cry in the airport, but he hugged her ... Then in the parking lot he started to cry like a baby," said Escobar's wife. "He has changed," said Escobar's sister Yunay. "His eyes shine now. He is very happy." Of how he was able to get his family out of Cuba, Escobar has said little else other than that his father and sister came to the United States "not through political or government connections, but through connections I had," Yunel Escobar said. "My mother came through different connections. They are all here and that's all I want to talk about. I feel that I've realized a dream," Escobar said. "Getting my family here, making my life complete."
The Braves drafted Escobar in the second round of the 2005 draft, 75th overall. "We saw him as a premium talent," says Roy Clark, Atlanta's scouting director. "A lot of clubs didn't feel that they had enough background [on him]."  Escobar's tense relationship with Cuban authorities in the aftermath of Peña's defection had limited his exposure to major league scouts.  With little more than a few off-season workouts to draw upon, most teams were reluctant to take a chance on Escobar. The Braves, however, spoke with Pena, who had been called up to the majors to play for the Braves only days before Escobar was drafted in the 2005 draft. Braves officials asked Pena countless questions about everything from Escobar's skill set to his command of English to his family background. "The best recommendation we got was from Brayan," Clark says. Escobar received a $475,000 signing bonus. 
After being drafted and signed by the Braves in 2005, Yunel began in the Appalachian League with Danville and proved that he was out of their league. He hit .400 with 2 homers and 8 RBIs in his short stay. He finished 2005 with Rome and was impressive there as well. In 198 at-bats, Yunel hit .313/.358/.470. He was progressing quickly just as Atlanta had hoped and expected from the 22-year-old first year player.
Escobar played the entire 2006 season in AA- Mississippi. His performance in Mississippi was somewhat disappointing with a few bright spots. The good news is that he showed good plate discipline with 59 walks and 77 strikeouts. This allowed him to post a very good .361 on-base percentage. However, he did not display any of the power that he showed in 2005. His slugging percentage dropped from .470 to .346 which caused his OPS to drop from 828 to 707. The source of his struggles in 2006 is hard to pinpoint but there were several things that didn’t work in Yunel’s favor. First, he did not get along with the manager, Jeff Blauser, at all. Secondly, playing two consecutive full minor league season probably put a toll on his body. He erased a lot of the doubt that people had in him by having an incredible season in the Arizona Fall League.
Escobar had a very strong spring training for the Braves in 2007 before being sent down to AAA Richmond to start the season. Escobar set the AAA International League on fire during his short stint there during the first two months of the 2007 season. In 46 games, Escobar hit .333 with a .379 OBP and a .456 SLG.
Escobar made his major league debut on June 2, 2007 against the Chicago Cubs at third base. He got a single in his first Major League at-bat, and finished the game 2-for-4 with a double, an RBI, and a run. His double in the top of the 8th inning gave the Braves the lead in their 5-3 win.
Escobar's first Major League home run came on June 4 against Florida Marlins pitcher Wes Obermueller. In the fourth inning of a 2-2 game - his first at Turner Field - Escobar took a high fastball from Obermueller the "opposite way", hitting it over the wall in right field. He finished the day 4-for-4 with two singles and a double in addition to his home run.
On July 27, in a game against the Arizona Diamondbacks, Escobar exhibited an uncanny sense of alertness (and riskiness) as the Braves were down 7-6 in the bottom of the ninth inning. With two outs, Escobar drew a walk from Arizona closer Jose Valverde. Seeing Valverde bending down with his back turned toward second base with no position player near the bag, Escobar sprinted immediately for second base. By the time Valverde realized what Escobar was attempting to do, it was too late to throw him out. He scored from second on the next play as Willie Harris hit a two-out single, tying the game at 7-7 and completing a seven-run comeback for Atlanta. The Braves, however, went on to lose in extra innings.
On August 5, Escobar recorded his first career walk-off hit against the Colorado Rockies in the 10th inning when he singled in Jeff Francoeur to give the Braves a 6-5 win. On August 29, Escobar belted his first leadoff homer in a 7-4 win over the Marlins. Escobar finished his very successful rookie season a very respectable .326 batting average.
Due in part to Escobar's emergence, the Braves traded starting shortstop Edgar Rentería to the Detroit Tigers following the 2007 season for Jair Jurrjens and Gorkys Hernandez. "We would have never traded Renteria if we didn't have someone like Escobar to follow," Braves manager Bobby Cox said. "He's an All-Star-caliber player." "He's a player," Cox said of Escobar, who hit .341 with a .408 on-base percentage from July 4 through the end of the season. "He can really play the game. Great hands, great arm, great range.
On June 2nd, 2008, exactly one year after his Major League debut, Escobar recorded his first career walk-off home run in a 7-5 victory over the Florida Marlins.
"For me, he could make the All-Star team," Braves manager Bobby Cox said of Escobar. "He has tremendous hands, range, and a tremendous arm. He's not going to steal a lot of bases, probably, but he can run. He can hit for average, and if he wants to, he can hit for power." 
"Escobar is one of the best players I've ever seen," Braves catcher Brian McCann said. "What he did last year was phenomenal. That wasn't a fluke. He's the real deal. He can make every play on the field. I think next year at this time, everybody will be talking about putting him in the class of [Florida shortstop] Hanley Ramirez and those guys." 
"I think he can be a .300 hitter in this league, no doubt in my mind," hitting coach Terry Pendleton said of Escobar. "He can be an Edgar Renteria, possibly with more power. He has a good idea of the strike zone, and he's aggressive in it. He might have to make adjustments this year as [pitchers] make adjustments to him, but he can hit. It showed last year, when he went against some teams a second and third time last year." 
"I've always had a lot of confidence in myself. It's just the game of baseball," said Escobar of himself.
“He’s going to hit for power,” Braves second baseman Kelly Johnson said of his double play partner Escobar. “I think it’ll be that right-center, (Marcus) Giles-back-in-the-day power. It’s gonna come. His [Escobar’s] hands … he’s got the best hands here. Stays inside the ball, consistently. He’s going to to add that pop.” 
"He's going to get to a whole lot of balls most guys won't get to, and he's going to make some throws other guys won't," said Braves first baseman Mark Teixeira, a former Gold Glove winner for Texas. "Because of that, there's going to be a few more errors."
“He’s a great kid. But in Cuba, they teach you how to hate the umpire, hate the other team, hate everybody on the field. When I was playing [in international tournaments] I’d see those guys in the hotel and they’d say, ‘Hey, we see you on TV, nice to meet you’ and all that. Then in the game you’d try to talk to them and the same guys would look at you like, ‘Get out of my face.’ “He’s a great kid. He’s learning and learning. Last night something happened, and he’s going to learn from that.” - Eddie Perez, Atlanta Braves coach speaking about Escobar's temper after Escobar was thrown out of a game for arguing with an umpire and firing his helmet to the ground in a game at Turner Field against the Milwaukee Brewers in June 2008.
“I’m a very intense player. I’m not trying to be nasty to the umpires. Sometimes it works both ways, there have been encounters [recently between players and umpires]. But I’m not a hot-tempered person, just intense.... I definitely am going to maintain that high level of [intensity] playing the gameand from this moment on I'm not even going to look at the umpires."” - Yunel Escobar, speaking in response to the debate about his intensity/temper.
"I just think Yunel plays with such passion, and has since the day we signed him," Braves general manager Frank Wren said of Escobar. "He's a very passionate player. He's exciteable, and he's exciting. All the things that ... make him fun to watch. At the same time, he's got to know when to temper that a little bit. That's the growing process. As he matures as a player, he's going to understand that more.... The umpires have their job to do and [Escobar] has got his job to do, and neither one of you should, in any way, try to make the other one look bad."
"He's trying, but sometimes in the heat of the game ... he's a passionate player.... It's a learning process. He's gotten better. He's trying, and we know that. He's not going to be perfect, but he's trying." - Braves coach Chino Cadahia speaking of Escobar's temper/intensity
- "Escobar enjoys memorable debut". MLB.com (2007-06-02).
- "Fatigued Hudson falls to Marlins". MLB.com (2007-06-05).
- "Big rally falls just short in the opener". MLB.com (2007-07-28).
- "Escobar plays hero for Braves.". MLB.com (2007-08-05).