The 29-year-old moves to Old Trafford on a five-year deal after United triggered a £35 million release clause in his contract.
"It's hard to describe just how proud I am to join Manchester United," Tielemans said. "Signing for such a special club feels incredible, it is the culmination of years of dedication since I first fell in love with football.
"I have had the privilege of experiencing success in the game and it has only increased my determination to achieve more.
"The ambition from everyone at the club is extremely clear; we are all determined to push for the biggest trophies over the coming years."
Tielemans spent three years at Villa after joining on a free transfer from Leicester and made 134 appearances.
He won the Europa League with them last season, scoring the opening goal in the final against Freiburg.
Before Villa, he won the FA Cup and Community Shield at Leicester.
Tielemans has 90 caps for Belgium and scored twice at this summer's World Cup as they reached the last 16.
United's director of football Jason Wilcox said: "Youri has consistently been one of the most outstanding midfielders in the Premier League throughout the past seven years. He has all of the technical qualities, as well as the ambition and mentality, to thrive at Manchester United.
"Youri's consistency is exceptional, and he will add further composure, creativity and leadership to our squad. We are delighted to welcome a player of his influence and experience, both on the pitch and within the dressing room, as we continue to build a team ready to challenge for the biggest honours."
Tielemans is United's third signing of the summer after midfielder Andrey Santos and goalkeeper Karl Darlow.
United acquired Santos from Chelsea in a £48 million deal, with £2 million in potential add-ons.
They also agreed a £35 million transfer for Atalanta midfielder Ederson, but the move collapsed over a medical issue.