The two clubs had previously reached an agreement "in principle" for the Welsh global, for a fee believed to be in the region of £18m.
James made 38 appearances for Swansea last season, scoring five goals and assisting another 10.
The Welshman turned heads for his terrifying speed, reaching 36 km/hr - just one km/hr shy of Arjen Robben's world record (37 km/hr).
Who could join Daniel James at Manchester United?
In signing for United the 21-year old has realised a childhood dream but it's mixed emotions for James as his father died suddenly aged 60 prior to the transfer going through.
The move was delayed as James had to link-up with the Wales squad ahead of their Euro 2020 qualifiers against Croatia and Hungary but with those games now out of the way, the youngster has been able to finalise his switch to Old Trafford.
He told the club's official website: "This is one of the best days of my life and a challenge I am really looking forward to". I think every kid grows up wanting to play for Manchester United.
"This is an incredibly proud moment for me and my family".
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Those performances netted him a move to Man United and Ole Gunnar Solskjær has praised James today.
"Daniel is an exciting young winger with lots of skills, vision, exceptional pace and a good work ethic", said the Manchester United manager. "He had a great season with Swansea City and has all the attributes needed to become a Manchester United player". The Premier League is the best league in the world and Manchester United is the flawless place for me to continue to develop as a player.
Despite some scepticism surrounding Manchester United looking towards the Championship to improve their squad, James' worldwide team-mate Joe Allen believes the 21-year-old is ready to make the step up. Thank you to everyone who has shown faith in me.
"I am confident he can do that".