The England international, who returned from the World Cup in Brazil just two days ago, is said to have attracted a bid from United on the final day of last season.
Southampton maintained that they would not allow their stars to leave the club until former manager Mauricio Pochettino's replacement was appointed.
Following on from Ronald Koeman's arrival at St Mary's, Shaw's switch to Old Trafford has been given the green light.
The 18-year-old, who is now the most expensive teenager in football, has penned a four-year contract, which has the option of a further year.
"I am thrilled and excited to be joining Manchester United," Shaw told the club's official website. "I have been at Southampton since I was eight years old and I would like to thank them and the fans for everything they have done for me.
"I want to continue to progress my career and joining United is the ideal place for me to do that. I am looking forward to this new chapter in my life and also looking forward to learning from the world-class players and management at the club."
Shaw made 67 appearances for Southampton following his debut in 2012.