Entering 2007, he was rated by Baseball America as the No. 2 prospect in the Atlanta Braves organization. In 2009, Andrus earned the Texas Rangers starting shortstop job at the age of 20.

Andrus was born in Maracay, Venezuela, the son of Emilio and Elvia Andrus. Emilio Andrus was a university professor; he died when Elvis was seven years old. He has two older brothers, Erold and Erikson. Some Venezuelan families name their children after famous non-Venezuelan people, but Andrus said that he is not sure whether he was named after Elvis Presley. Andrus said, "I asked my mother a lot of times, but she never told me. Maybe my dad liked Elvis Presley. I’m not sure about that."

At the age of 16, Andrus was signed by the Atlanta Braves and received a signing bonus of more than $500,000. He started the 2005 season with the Orlando Braves of the Gulf Coast League, and was promoted to Danville late in the season. He spent 2006 with Class-A Rome, being named to the South Atlantic League All-Star team.

In 2007, Andrus joined the Myrtle Beach Pelicans of the Carolina League. During the season he was selected to the All-Star Futures Game and was selected again for the showcase in 2008.

On July 31, 2007, Andrus, Jarrod Saltalamacchia and three minor league pitchers (Matt Harrison, Neftalí Feliz and Beau Jones) were traded from the Braves to the Rangers for Mark Teixeira and Ron Mahay.

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