Escobar was born in Havana, Cuba. Brayan Peña, 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 and sharing bats, gloves, and cleats.

In 2004, Escobar defected from Cuba by sea, and landed in the Florida Keys.

The Braves drafted Escobar in the second round of the 2005 draft, 75th overall. "We saw him as a premium talent", said Roy Clark, Atlanta's scouting director. "A lot of clubs didn't feel that they had enough background [on him]." Former Braves general manager John Schuerholz said he does not hesitate when he finds a Cuban who fits his team's needs. "I'm not a politician", Schuerholz said. "I'm a baseball general manager. We look for baseball talent wherever we can find it. And wherever it is, we go after it." 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.

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