The 21-year-old has penned a four-year deal with the Blades, keeping him at the club until 2027.
The two clubs were close to agreeing a £18 million fee for the striker but Villa inserted a buyback option into the deal.
United recently lost Iliman Ndiaye to Marseille and have signed Cameron as a replacement for him.
The young attacker came through the youth ranks at Villa, earning his competitive debut at the age of 17 in 2019.
He made 14 appearances in all competitions for the Midlands outfit but spent most of his time on loan to other clubs.
The centre-forward was at Preston North End for the second half of the 2021-22 season.
He scored seven goals in 20 Championship appearances for Preston before going on loan to Middlesbrough the following campaign.
The English footballer played an instrumental role in helping Boro make it to the play-offs, where they lost to Coventry City in the semi-final.
He registered 11 goals and six assists in 22 league appearances, averaging 1.2 key passes, 1.8 shots and 1.1 dribbles per game.
Archer has now joined United who finished second in the Championship to return to the Premier League.
The 21-year-old is their eighth signing of the 2023 transfer window.
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