The 33-year-old, who signed a new one-year contract with United in May, has committed to a two-year deal with the Serie A champions.
Evra's departure leaves £30 million signing Luke Shaw as the only specialist left-back in United's first-team squad, after Alexander Buttner moved to Dynamo Moscow.
On the announcement of the move, Evra told United's official website: "After a great deal of thought I have decided the time is right for me to leave Manchester United.
"It is the biggest decision of my career as this club is, and will always remain, in my heart."
Juventus have revealed they will pay an initial fee of £1.2m in two installments, followed by an additional £300,000 should they finish in the Champions League qualification places next season.
Evra is the fourth decorated United veteran to leave the squad this summer, after Nemanja Vidic and Rio Ferdinand signed for Inter and QPR respectively and Ryan Giggs announced his retirement from the game.
The Frenchman made 379 appearances in a trophy-laden nine-season spell at Old Trafford, scoring 10 goals.