James, who put pen to paper on a five-year contract with the option for a further season, becomes the club’s first summer signing.
The transfer completes a meteoric rise for the Wales international following his breakthrough campaign at Championship club Swansea.
“This is one of the best days of my life and a challenge I am really looking forward to,” James said.
“The Premier League is the best league in the world and Manchester United is the perfect place for me to continue to develop as a player.
“This is an incredibly proud moment for me and my family, however, it is tinged with sadness that my father isn’t able to share this experience with us.
“I am looking forward to working with Ole, meeting the team and getting started.”
James, who made his senior debut for Swansea in February 2018 after a nightmare loan spell with Shrewsbury Town, provided five goals and 10 assists in 38 appearances in all competitions for Swansea last season.
The 21-year-old made his international debut last November and has won four caps for his country.
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He attracted plenty of attention after running the full length of the pitch to score against Brentford in the FA Cup fifth round in February.
And James tormented Manchester City in the FA Cup quarter-finals, which saw Pep Guardiola hail his performances.
Despite his relative inexperience, Ole Gunnar Solskjaer is convinced James will be an instant smash hit at Old Trafford.
And Solskjaer revealed the Red Devils pursued his signature because of his speed and wing play.
“Daniel is an exciting young winger with lots of skills, vision, exceptional pace and a good work ethic,” he said.
“He had a great season with Swansea City and has all the attributes needed to become a Manchester United player.
“We are delighted he has signed with our club and we are all looking forward to working with him.
“This is the perfect environment for Daniel to continue his development.”