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 the two clubs previously agreed a £73million deal.
Solskjaer has finally got his man 12 months after first registering his interest only for United to be put off by the Bundesliga club insisting on a £100m fee.
“The chance to join Manchester United is a dream come true and I just cannot wait to perform in the Premier League.
“This is a young and exciting squad and I know, together, we can develop into something special to bring the success that the fans deserve.
“I am looking forward to working with the manager and his coaching team to further develop my game.”