The Spain international flew north on Saturday and successfully came through a medical before finalising the biggest move to date of the current January window.
Mata will train with his new-team mates in the coming days and is expected to make his debut against Cardiff at Old Trafford on Tuesday night.
He said: "I am thrilled to be joining United. 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.
"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."
The transfer, which far exceeds United's £30.75m deal for Dimitar Berbatov back in 2008, represents a real coup for David Moyes following a summer of frustration in the market and a faltering defence of United's Premier League title in his first season at the club.
He said: "Juan is one of the finest playmakers in the game today and it's a real pleasure to have secured his signature.
"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."
Mata was Chelsea's player of the season in each of the past two years but had been reduced to a bit-part role under Jose Mourinho.
His departure has not gone down well with some Chelsea supporters, but the deal represents a hefty profit on a player they signed for £23.5m from Valencia in 2011.
Mourinho, who has lined up Basel's Mohamed Salah as a replacement, said on Saturday: "We want him to be successful in his career and as well as that, this transfer makes huge financial sense to us at Chelsea.
"We are pleased Juan will stay in England - even with a rival club. He is happy in this country and it is good the Premier League retains such a talented player."
Mata, meanwhile, published a lengthy open letter on his website thanking Chelsea fans for their support.
The 25-year-old said he planned to stay "for a long time" before falling out of favour, although there was no criticism of Mourinho.
He said: "Obviously, I fully respect the circumstances. Football is a team sport and it's not easy to play regularly in such a good squad."
Mata even went on to thank Mourinho "for making me a stronger player, because in difficult moments, I have learnt to remain positive and to keep working hard."