The striker has signed a five-year deal with the Italian champions, though los Blancos retain a buyback option of up to €30m [23.7m], and says he is "itching" to begin Serie A life
Morata arrived in Turin on Friday ahead of a medical with the Italian champions after weeks of speculation over a possible summer switch.
Juve have now confirmed the 21-year-old has agreed a five-year deal, and have also revealed that Madrid will retain a buy-back option in the player's contract which could rise to €30m [£23.7m].
"Juventus announce that it has reached an agreement with Real Madrid for the definitive purchase of the rights to the player Alvaro Borja Morata Martin for a fee of €20m, payable over three years," a statement on their official website read.
"The agreement also provides an option for Real Madrid, exercisable at the end of the 2015-16 season or 2016-17 season, for the outright repurchase of Morata, up to a maximum of €30m depending on the number of games the player has played for Juventus.
"Juventus have signed a five-year contract of employment with the same player."
Morata, who scored eight goals in 23 Liga appearances for Madrid last season, admitted in his press conference he has been itching to make the move for some time and is relishing the prospect of life under new boss Massimiliano Allegri.
"Juve are one of the best clubs in the world and Italy's finest. I'll try to contribute to new successes," he said.
"I've spoken to Allegri and I'm itching to get down to work. I've wanted to come to Juventus for a long time. The club's always shown a strong interest in me.
"I feel prepared for the new season ahead, I know it's a challenging league but I'm ready to tackle Serie A."
Morata becomes the second player involved in a transfer away from Madrid on the same day, with midfielder Casemiro's loan move to Porto also confirmed on Saturday.