Van Persie has signed a four-year contract with the Red Devils.
United boss Sir Alex Ferguson told his club's official website: "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's an honour to sign for Manchester United.
"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."