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 United have strengthened their goalkeeping options by bringing in experienced Wales international Karl Darlow on a two-year deal, with the agreement including the option of an additional 12 months.
Brighton have completed the signing of Croatia defender Luka Vuskovic from Tottenham Hotspur in a club-record deal worth up to 50 million pounds.
Liverpool manager Andoni Iraola has moved to ease uncertainty surrounding Harvey Elliott, delivering his first public comments on the midfielder's future since taking over at Anfield.
Manchester United have confirmed the signing of Andrey Santos from Chelsea, while the Red Devils are also closing in on a deal for the transfer of Youri Tielemans from Aston Villa.
