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He replaces Dutchman Louis van Gaal, who was dismissed on Monday, two days after the club's FA Cup win.
"Jose is quite simply the best manager in the game today," said United executive vice-chairman Ed Woodward.
Mourinho, 53, said: "To become Manchester United manager is a special honour in the game. It is a club known and admired throughout the world."
He said he had always "felt an affinity with Old Trafford" and claimed he has "always enjoyed a rapport with the United fans".
He added: "I'm looking forward to being their manager and enjoying their magnificent support in the coming years.
"There is a mystique and a romance about it which no other club can match."
In an interview with MUTV, Mourinho said he is "in the right moment" to succeed at United, adding the "past three years" did not reflect the club's history.
Saturday's win brought United their first piece of silverware since the retirement of Sir Alex Ferguson in 2013.
"It is a giant club. It must be for the best managers," he said. "I prefer to forget the past three years and focus on the giant club I have in my hands.
"I want to win. What the players need to do is listen. I want to focus on the history of this giant club. I will give everything to go in the direction we all want."
"The most important thing is the players and my relationship with them."
Mourinho said United fans had shown him "empathy, no problems" on previous visits.
"I was pushed by that feeling to say things my clubs were not happy with, like when I said the best team lost when I won with Real Madrid," he added.
"Real Madrid were not happy with that."
The Portuguese's appointment followed three days of talks between his agent, Jorge Mendes, and senior United officials.
He has already been linked with a move for Sweden striker Zlatan Ibrahimovic, who played for him at Inter Milan.
Ibrahimovic, who has decided to leave Paris St-Germain, told reporters on Thursday that Mourinho "is the man" to bring United "back to the top".
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The BBC reported on Saturday, within an hour of United's 2-1 win over Crystal Palace, that Van Gaal would be sacked and replaced by Mourinho.
Talks, which started on Tuesday and were delayed by negotiations over Mourinho's image rights, finally concluded on Thursday.
As well as his three Premier Leagues, Mourinho has led Real Madrid to the Spanish title in 2012, and guided Porto and Inter Milan to Champions League victories in 2004 and 2010 respectively.
However, he left the Blues in turmoil only seven months after winning the Premier League title, with the club 16th in the league, just one point above the relegation zone, having accused some players of "betraying" his work.
Mourinho's first match at Old Trafford will be on 5 June for Soccer Aid, managing an England team against a Rest of the World side led by Leicester boss Claudio Ranieri.
He will then come up against Guardiola, who takes charge of Manchester City, during a pre-season tournament in Beijing on 25 July.
Mourinho was manager of Real Madrid from 2010 to 2013, winning one La Liga title, while Guardiola was coach of Barcelona.
"We will undoubtedly have fun," journalist Graham Hunter told BBC Radio 5 live. "I hope we get the quality of football, drama, theatre that we got then. A very different situation coming your way in Manchester next season."
The future of assistant boss Ryan Giggs remains uncertain, with the club not mentioning the 42-year-old in their statement.
He is known to be unhappy at the way Van Gaal was dismissed and is considering an offer to extend his 29-year association with the club.
He has now passed over for the manager's job for the second time.
Former Chelsea winger Pat Nevin:
"From the very start of his career in coaching he has had this incredible self-belief, great knowledge and incredible attention to detail, but also this ability to think outside the box in a way some managers don't.
"He's very willing to change things and adapt things in more complex ways when he has to and when it has to be simple, he will be simple.
"I have talked to him a few times about tactics. This guy is on a different level completely. In one game he made a number of changes within a three-minute period that other managers would take three games to do.
"He made radical changes that would not have even entered the mind of most managers. He not only did it, he did it with speed and they worked."
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Jose Mourinho has been officially confirmed as Manchester United manager and has signed a three-year contract.