The 23-year-old midfielder returns after four years away, having left United for Juventus for £1.5m in 2012.
Pogba, who signed a five-year deal, added: "This is the right club for me to achieve everything I hope to in the game."
Jose Mourinho said Pogba would be "the heart of the club" for the next decade.
United will pay the Italian champions 105m euros for Pogba, plus performance-related bonuses and other costs.
It surpasses Wales forward Gareth Bale's £85m move to Real Madrid in 2013 as the biggest transfer deal in history.