United met a release clause of £35m for the 29-year-old midfielder, who helped Belgium reach the quarter-finals at the FIFA World Cup before losing out to Spain.
Tielemans had been at Villa for the last three seasons, helping the English club win the Europa League against German side Freiburg in May.
A former team-mate of United's current first-team coach Jonny Evans, Tielemans is noted for scoring Leicester's winner against Chelsea in the 2021 FA Cup final.
He said: "It's hard to describe just how proud I am to join Manchester United. 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."
Jason Wilcox, director of football, 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."
Manchester City are stepping up their pursuit of Moroccan teenager Ayyoub Bouaddi, with reports claiming the Premier League champions are preparing a 90 million euros bid for the Lille midfielder.
Manchester City have secured the long-term future of defender Abdukodir Khusanov after the Uzbekistan international signed a new five-year contract with the Premier League club.
Manchester United claimed their first pre-season win on Friday with a commanding 5-0 triumph over Rosenborg at Lerkendal Stadion, prompting manager Michael Carrick to single out the club's emerging talent.
Jeremy Jacquet is settling quickly into life at Liverpool after his big-money move from Rennes.
