The France left-back, 33, has signed a two-year contract with the Italian champions, ending his eight-and-a-half-year stint at Old Trafford.
"Playing for Manchester United has been a privilege, a dream come true, and my time there has been so memorable," he said.
Evra joined United in January 2006 from Monaco for £5.5m and won five Premier League titles.
He signed a one-year extension to his contract in May but asked to leave when Juventus approached United over his availability.
Evra would have been competing for a starting berth next season against £27m signing Luke Shaw.
"After a great deal of thought I have decided the time is right for me to leave," Evra said.
"It is the biggest decision of my career as this club is, and will always remain, in my heart."
Evra made 379 appearances for United and as well as Premier League success, he helped the club win the Champions League in 2008, three League Cups and the Club World Cup.
He was part of the France squad at the 2014 Fifa World Cup squad in Brazil and played in four out of their five games, including the quarter-final defeat by Germany.
As he departed, Evra also thanked former manager Sir Alex Ferguson and the club's fans.
"An immense thank you to Sir Alex for making it all possible, for giving me the privilege to be a captain and to understand that nobody is bigger than the club," he said.
"My biggest thanks go to the fans. We have had some fantastic times together and every single time I stepped out in that shirt I knew I was playing for a very special club."
Juventus will pay an initial £1.2m for Evra, plus an extra £300,000 if they qualify for the Champions League in 2015-16.
Red Devils executive vice-chairman Ed Woodward has told United fans to expect more signings over the next few weeks as new manager Louis van Gaal attempts to build a side capable of competing for the Premier League title.
Patrice Evra has completed a £1.2m move to Juventus from Manchester United.