Onana has agreed a five-year deal with the Premier League club and will succeed David de Gea (32) as number one at Old Trafford.
United are set to pay an initial £44million for the keeper with a further £3.5million due in performance-related add-ons.
"To join Manchester United is an incredible honour and I have worked hard all my life to get to this moment, overcoming many obstacles along the way," Onana said in a United statement.
"Walking out at Old Trafford to defend our goal and contribute to the team will be another amazing experience. This is the start of a new journey for me, with new teammates, and new ambitions to fight for."
De Gea left the Red Devils earlier this month after 12 years at United.
The Spaniard struggled with Erik ten Hag's preference to play out from the back and Onana appears a far more natural fit for the Dutch coach's philosophy.
Onana joined Barcelona as a 14-year-old after being spotted at Samuel Eto'o's academy in his homeland.
After failing to make the first team at the Camp Nou, Onana joined Ajax in 2015 and won three Eredivisie titles under Ten Hag in the Netherlands.
However, his career was temporarily rocked in 2021 by a doping ban for taking a banned diuretic.
Onana claimed to have accidentally taken a drug prescribed to his wife and had a one-year ban reduced to nine months on appeal at the Court of Arbitration for Sport.
He joined Inter on a free transfer just over 12 months ago when his contract at Ajax expired.
Onana made only 24 Serie A appearances as he shared goalkeeping duties with Samir Handanovic (39) in the Inter captain's final year at the club.
But he was an ever-present in the Champions League, keeping eight clean sheets in 13 games as Inter reached the final for the first time in 13 years.
Onana played for Cameroon in their first match at last year's World Cup, but he was dropped for a disciplinary issue the rest of the tournament in Qatar and subsequently announced his international retirement.
He is United's second close-season signing after England midfielder Mason Mount (24) arrived from Chelsea earlier in July for an initial £55 million.
Onana could make his first appearance for United in their friendly against Arsenal on Saturday in New Jersey.
Ten Hag is also keen to sign a forward before the start of the new Premier League campaign.
The United boss, who ended his club's six-year trophy drought by winning the League Cup last season, has been linked with a bid for Atalanta's Denmark striker Rasmus Hojlund (20).
