United had already confirmed on Friday that they had agreed a deal to sign the 25-year-old for a club-record fee.
"Today is a very happy day... It's a massive club and I feel very proud to be here," Mata told manutd.com.
"I am thrilled to be joining. United is the perfect place for me."
United manager David Moyes described Mata as "one of the finest playmakers in the game".
"It's a real pleasure to have secured his signature," Moyes said.
The deal eclipsed the £30.75m that United paid Tottenham Hotspur in 2008 to sign Bulgarian striker Dimitar Berbatov.
Mata joined Chelsea from Valencia in 2011 and was voted the club's player of the year in his first two seasons, but he has fallen out of favour since Jose Mourinho returned to Stamford Bridge as manager last year.
Mata is expected to go straight into the squad for United's home game against Cardiff City in the league on Tuesday.
His signing reflects Moyes's eagerness to arrest a startling decline in United's fortunes since he succeeded fellow-Scot Alex Ferguson as manager at the end of last season.
United are currently 14 points behind leaders Arsenal in the Premier League and have been knocked out of both the FA Cup and the League Cup this month, although they have reached the last 16 in the UEFA Champions League.
United striker Robin van Persie welcomed Mata's arrival, writing on Twitter: "Welcome Juan Mata see you all tuesday."
Centre-back Rio Ferdinand tweeted: "Welcome to my amigo @juanmata10! Hoping you enjoy ya time here and help continue the success at this club."
Mata will join countryman David de Gea at Old Trafford and the United goalkeeper gave his fellow Spaniard a playful welcome, tweeting: "Welcome to my brother @juanmata10 just one thing....Not the best clothes!"
Meanwhile, reports in the British media on Saturday claimed that United have opened talks with striker Wayne Rooney over a new contract.