The Spain midfielder, 25, had a medical on Saturday and becomes manager David Moyes's second high-profile signing, following the arrival in August of Everton's Marouane Fellaini for £27.5m.
Mata is likely to make his United debut at home against Cardiff on Tuesday.
"The time has come for a new challenge," he said. "United is the perfect place for me."
Mata added: "I am excited at the chance I have to be part of the next phase in the club's history.
"Chelsea is a top club and I have many friends there but you cannot turn down the chance to join Manchester United. I look forward to helping the manager and the team be successful in the coming years."
Mata, who joined Chelsea from Valencia for £23.5m in August 2011, has been Chelsea's player of the year in each of the last two seasons.
He scored 32 goals for the Blues and made over 130 appearances but has failed to command a regular starting place under Jose Mourinho, who returned to Stamford Bridge last summer for a second spell as manager.
"Juan is one of the finest playmakers in the game today and it's a real pleasure to have secured his signature," said Moyes.
"He's been instrumental in Chelsea's recent success, with a notable 28 assists and 32 goals in his last two Premier League seasons.
"I have been impressed with the excitement he has shown to be part of the new Manchester United."
"I know our fans have admired Juan for some time now - he has always played well against us - and I know they will be delighted to see him in the red of United, starting on Tuesday."
Welcum bro.... But be rest assure yu are not going to the champions leaque next season....