The 21-year-old England international – a former Manchester City youth team player – has signed a five-year contract, with the option of a further year, after a £73million deal was previously agreed with Borussia Dortmund.
United's last trophy was the Europa League in 2017, while their last domestic title success was eight years ago in Sir Alex Ferguson's final season as manager.
"He's got that DNA of exciting fans and entertaining and he's also a winner. He wants to win and that's the next step for us.
"We need to take the next step, we want to take the next step and we feel that he can help us do that.
"It's Man United and we know that our fans want us to go one step further than we did last season, and we're looking forward to the challenge."