The French midfielder, 19, turned down a contract offer from the Reds, who revealed last month that he had agreed to join the Italian champions.
"I am very happy to be here, it is a real honour," he told Juventus' official website.
"I have been told that there is a very good standard of play and the fans support their favourites.
"I hope they take me and support me in the same way."
Pogba was criticised by United for engineering the free transfer move, although he was signed by the club in similar circumstances in 2009.