The 19-year-old has penned a long-term contract and could cost United significantly more than the initial £36million fee with numerous add ons involved in his Old Trafford deal.
Martial, who had been called up to the France squad for the first time, left their training base at Clairefontaine on Monday to undergo a medical at Old Trafford and finalise his move.
The former Lyon striker, who joined Monaco for 5m euros in 2013, has scored 15 goals in 72 games in his senior career.
“I’m so excited to join United,” Martial told the club’s website. “I’ve always wanted to play in the Premier League and look forward to meeting my team-mates.”