The 23-year-old midfielder returns after four years away, having left United for Juventus for £1.5m in 2012.
Pogba, who has signed a five-year deal, added: “This is the right club for me to achieve everything I hope to.”
Manager Jose Mourinho said the France international could be “the heart of the club” for the next decade.
United will pay the Italian champions 105m euros for Pogba, plus five million euros (£4.5m) in performance-related bonuses, and other costs, including five million euros if Pogba signs a new contract.
The fee surpasses the £85m Real Madrid paid Tottenham for Wales forward Gareth Bale in 2013.
The announcement was made by United just after 00:30 BST on Tuesday, witha glossy video showing the player – with a red devil marked into his hair – emerging from the shadows.