Pogba, 23, returns to Old Trafford four seasons after departing for Italy, with Juventus confirming the transfer fee of €105 million (£89.3 million, $116.4 million).
The fee, payable over the next two years, can increase by an additional €5 million if certain conditions are met, Juventus said.
The transfer beats the previous record fee of an estimated €100 million (£85 million) that Real Madrid paid Tottenham for Gareth Bale in 2013.
When Pogba joined Juventus in 2012, United received just £800,000 in compensation, and the Frenchman quickly established himself as a key player, helping the Bianconeri win Serie A four times and the Coppa Italia twice.
Pogba had made just seven appearances for United during his first spell with the Premier League club, all during the 2011-12 season, including three in the Premier League, three in the Capital One Cup and one in the Europa League.
All seven of those were second-half substitute appearances, and he opted to leave when his contract expired after failing to establish himself in Sir Alex Ferguson’s first-team plans.
He returns to United on a five-year contract, with an option to extend the deal for an additional season.
“I am delighted to rejoin United,” Pogba told United’s official website. “It has always been a club with a special place in my heart and I am really looking forward to working with Jose Mourinho.
“I have thoroughly enjoyed my time at Juventus and have some fantastic memories of a great club with players that I count as friends.
“But I feel the time is right to go back to Old Trafford. I always enjoyed playing in front of the fans and can’t wait to make my contribution to the team. This is the right club for me to achieve everything I hope to in the game.”
Pogba arrived at Manchester United’s training ground on Monday to have his medical exam, before the team confirmed the deal just after midnight on Tuesday.
Pogba’s agent, Mino Raiola — who, sources told ESPN FC, is set to make €20 million from the deal — appeared to confirm the transfer was done shortly before United published their announcement.
Real Madrid had also been keen to sign Pogba, and sources told ESPN FC the European champions made an attempt last month to beat United to the deal.
Instead, he becomes Mourinho’s fourth signing since taking charge this summer, following Eric Bailly, Zlatan Ibrahimovic and Henrikh Mkhitaryan.
“Paul is one of the best players in the world and will be a key part of the United team I want to build here for the future,” Mourinho said in the club’s statement.
“He is quick, strong, scores goals and reads the game better than many players much older than he is.
“At 23, he has the chance to make that position his own here over many years. He is young and will continue to improve; he has the chance to be at the heart of this club for the next decade and beyond.”
The opportunity to play in the Premier League will be important for Pogba, Mourinho said after Sunday’s Community Shield victory over Leicester.
“United is the perfect club to bring him to the level he wants to be,” the manager said. “The Premier League is the perfect habitat for that. If you want to be the best player in the world, if you go to Barcelona or Real Madrid, you are in trouble because I don’t think the other two big guys [Lionel Messi and Cristiano Ronaldo] will let you.”