Van Persie agreed to personal terms and passed a medical examination to complete a transfer that is believed to have cost United about 24 million pounds ($38 million).
"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," Van Persie said. "I can't wait to get started."
While Arsenal will be happy to have recouped such a big fee for a 29-year-old player who has had many injuries in his eight years at the club, Van Persie's departure deprives manager Arsene Wenger of his captain and the league's top scorer last season with 30 goals.
In fact, Van Persie becomes the fifth Arsenal captain to leave the club since 2004, after Patrick Vieira, Thierry Henry, William Gallas and, in the preseason of 2011, Cesc Fabregas.
Van Persie's future at Arsenal has looked bleak ever since he issued a statement on his website last month saying he disagreed with the direction the club was heading. The London team hasn't won a trophy since 2005, when it beat United on penalties to capture the FA Cup.
Manchester City and Juventus initially showed interest in Van Persie, but Ferguson maintained his pursuit of the Dutchman, who could now make his debut against Everton in United's Premier League opener at Goodison Park on Monday.
"Robin is a world-class striker with a proven record in England and in European football," United manager Alex Ferguson said. "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's departure provides further evidence that Arsenal can no longer compete with the likes of Manchester City, United and Chelsea.
Van Persie will join an attacking unit at Old Trafford that already includes Wayne Rooney, Danny Welbeck and Javier Hernandez. Dimitar Berbatov is also still at United, but is likely to leave to make way for Van Persie.