The 18-year-old has signed a four-year contract, with an option for a further 12 months, at Old Trafford having secured the move after returning from World Cup duty with England.
United had an initial bid of £27m for Shaw turned down at the end of last season but an improved offer sealed the deal.
The teenager, who made a single appearance in the World Cup, joins Ander Herrera in moving to Manchester; the midfielder is moving on a four-year contract from Athletic Bilbao and for a fee of around £28m.
Shaw said: "I am thrilled and excited to be joining Manchester United. 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."
Manchester United assistant manager Ryan Giggs, assistant to Louis van Gaal who is in charge of Holland in Brazil, said: "Luke is a very talented young left-back with great potential.
"He has developed immensely during his time at Southampton and has all the attributes to become a top player. I am delighted he has joined Manchester United as I believe this is the right club for him to continue his development as a young player and eventually fulfil that promise."
Veteran Patrice Evra signed a one-year contract extension in May but United had been keen to secure a long-term replacement for the Frenchman, especially after Alex Buttner agreed to join Dinamo Moscow earlier this week.
Shaw and Herrera appear to be just the start of van Gaal's rebuilding job, as the Dutchman looks to banish the memory of last season's seventh-placed finish.
Shaw will be competing with Evra for the left-back role at United, but is understood to have been told he will be first choice despite competition from the France international and not turning 19 until July 12.
Shaw, who becomes the world’s most expensive teenager as a result of the transfer, made his Southampton debut as a 16-year-old in January 2012.
He has played 60 Premier League games for Saints and earned his first England call-up for the friendly against Denmark in March.
Roy Hodgson selected Shaw ahead of Ashley Cole for England’s World Cup squad, but he only made one appearance - in the final game against Costa Rica.
A day later he was back on English soil, where the wheels quickly moved into motion for his move to United, undergoing a medical on Thursday and signing the following day.
Shaw is the latest high-profile departure from Southampton, who saw manager Mauricio Pochettino leave for Tottenham last month.
That exit was swiftly followed by talisman Rickie Lambert moving to boyhood club Liverpool, who are now poised to sign Adam Lallana - an interesting headache for recently-appointed manager Ronald Koeman.