United confirmed they had agreed a fee with Arsenal on Wednesday evening, and after a medical on Thursday the final elements of the transfer have been wrapped up.
The deal has been done in time for Van Persie to go straight into Sir Alex Ferguson's squad for Monday's trip to Everton.
United announced in a statement: "Robin van Persie has completed his medical, agreed personal terms and signed for Manchester United on a four-year contract until June 2016."
Manager Ferguson said: "Robin is a world-class striker with a proven record in England and in European football.
"His talents need no introduction to our fans - he has scored a number of goals against us in some classic battles with his former club.
"His movement, finishing and all-round ability are outstanding."
Van Persie said it was an "honour" to sign for United.
He said: "I am looking forward to following in the footsteps of so many great strikers, bringing my experience and playing my part to help the team compete for the biggest trophies in the game. I can't wait to get started."
Van Persie has signed a four-year contract with United, bringing to an end one of the longest-running transfer sagas of the summer.
Ferguson has fought off competition from Manchester City and Juventus to land Van Persie.
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However, he also needed to break an impasse with Arsenal boss Arsene Wenger, who did not want to sell last season's Golden Boot winner to a rival club.
Even a week ago, Wenger had not given up hope of persuading the 29-year-old to stay.
However, when Van Persie was booed by Arsenal fans at a pre-season game in Germany on Sunday, it signalled the end of his Gunners career.
Ferguson told MUTV: "The boy wanted to come. He turned down various clubs. If he hadn't told Arsenal he wanted to go to Manchester United, the transfer wouldn't have happened."
Ferguson admitted he thought it would be impossible to get Van Persie to sign, but had to make an attempt when the striker did not sign a new contract with the north London club.
"It has given everyone a lift," said Ferguson.
"I have a complication to my life (selection). I don't know how I managed in 1999.
"They can gel together. Robin's best position is furthest up the pitch. The most important thing is to win what we want to win. These players can help us to do that."
Ferguson revealed he was still looking at a new signing for "one more position" but "only if a certain player came available".