Andre Onana played under United manager Erik ten Hag at Ajax, winning three league titles
The Cameroon international, 27, has signed a five-year contract, with the option of a 12-month extension.
The deal is worth an initial £43.8m plus a possible £3.4m in add-ons, and the former Ajax keeper is set to join up with the United squad on their pre-season tour of the United States.
"To join Manchester United is an incredible honour," he said.
"I have worked hard all my life to get to this moment, overcoming many obstacles along the way."
Onana will replace David de Gea as United's first-choice keeper, with the Spaniard having left in July when his contract expired after 12 years at the club.
"Walking out at Old Trafford to defend our goal and contribute to the team will be another amazing experience," Onana added.
"This is the start of a new journey for me, with new team-mates and new ambitions to fight for.
"Manchester United has a long history of incredible goalkeepers, and I will now give everything to create my own legacy in the coming years."
Onana played under United manager Erik ten Hag during a seven-and-a-half-year spell with Ajax, winning three league titles, and served a nine-month ban in 2021 for a doping violation.
After joining Inter on a free transfer in July 2022, Onana helped the Italians reach last season's Champions League final, where they lost 1-0 to Premier League champions Manchester City.
"I am excited by the opportunity to work again with Erik ten Hag, and I can't wait to play my part in the success I know he is determined to deliver at this great football club," Onana said.
He kept eight clean sheets in 24 Serie A games as Inter finished third, plus 13 in the Champions League last season - the most of any goalkeeper - giving him a total of 104 clean sheets in 255 club appearances.
Shortly before his move was announced, Onana posted a farewell message to Inter on social media saying: "I don't know if I was the best, but I always gave my best."
United's football director John Murtough added: "Andre was our top choice to become the new goalkeeper for Manchester United, with exactly the right profile of technical attributes and personality.
"Having achieved success throughout his career, we know that Andre will further add to the winning mentality we are building within our squad.
"He is already one of the world's best goalkeepers, and at the age of 27, and we firmly believe that he can develop even further over the coming years."
Onana retired from international football in December after falling out with Cameroon coach Rigobert Song at the World Cup.
-- Courtesy of BBC Sport