Van Persie agreed 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 injury problems in his eight years at the club, Van Persie's departure nevertheless 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 personal 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.
"Let me make one thing clear, I don't want things to get twisted," Van Persie said. "From my side and Arsenal's side now, there are no hard feelings. It's just certain elements are vital for me and we had different views."
The Netherlands international also spoke of his excitement at playing alongside players such as Wayne Rooney, Paul Scholes and Ryan Giggs.
"Of course I'd love to play with Wayne," Van Persie said. "''I've said many times before he is a world-class player all-round.
"Scholes and Giggs are world class and have been proving it for 20 years now. That had a big impact (in my decision), definitely. I'd like to say we are all in it together to achieve greatness."
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.
Van Persie's departure provides further evidence that Arsenal can no longer compete with the likes of City, United and Chelsea.
In the past three years, Samir Nasri, Gael Clichy, Emmanuel Adebayor and Kolo Toure have all joined big-spending City, which won the English title last season. Fabregas left to join Barcelona, widely seen as the best team in world football, and now a disillusioned Van Persie has departed.
Wenger has already brought in Olivier Giroud and Lukas Podolski in anticipation of Van Persie being sold, meaning there is no need for the last-minute signings that happened on transfer deadline day last August after the reluctant sales of Nasri and Fabregas.
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.
He is United's third signing of the preseason, after Japan playmaker Shinji Kagawa from Borussia Dortmund and midfielder Nick Powell from English lower-league team Crewe Alexandra.
"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."