"I am thrilled to be joining United," the 25-year-old told the club's website (www.manutd.com) on Saturday.
"I have enjoyed some very happy years at Chelsea but the time has come for a new challenge. United is the perfect place for me and I am excited at the chance I have to be part of the next phase in the club's history."
Mata claimed the Player of the Year award in his first two seasons at Chelsea, helping the Londoners win the 2012 Champions League and 2013 Europa League, but has been out of favour this season under manager Jose Mourinho.
"Juan is one of the finest playmakers in the game today and it's a real pleasure to have secured his signature," United manager David Moyes said.
"He's been instrumental in Chelsea's recent success with a notable 28 assists and 32 goals in his last two Premier League seasons."
The fee breaks the previous record of 30.75 million pounds United paid Tottenham Hotspur for Bulgaria striker Dimitar Berbatov in 2008.
($1 = 0.6060 British pounds)
(Reporting by Josh Reich; editing by Tony Jimenez)
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