Van Persie, 29, is believed to have cost United £24 million and has joined the club in time to make his debut against Everton on Monday night.
It is reported he will earn £250,000 per week at Old Trafford.
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.”
The Netherlands international forced Arsenal into a sale after publicly stating he would not extend his contract, which had only one year left to run at Emirates Stadium.
He joined Arsenal from Feyenoord in a £2.75 million deal in May 2004 and scored 132 goals for the Gunners, winning the FA Cup when beating United in the final in 2005.
He was made Arsenal captain last summer following the sale of Cesc Fabregas to Barcelona but now completes a formidable United attack that also includes Wayne Rooney, Danny Welbeck and Javier Hernandez.
Sir Alex 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.”
Arsenal have appointed Thomas Vermaelen as their new captain and Arsene Wenger said on Friday that the club have already compensated for the loss of the Dutchman in attack.
“We regret that we have lost a great player and for one simple reason - he only had one year left on his contract. He wanted to go,” Wenger said.
“We anticipated this situation might happen, therefore we bought Lukas Podolski and Olivier Giroud.
“[Selling to a rival] is not what we wanted but there was not much choice. We had only one club really interested at the level we thought would be acceptable and that was Manchester United.
“At the end of the day, [selling] was the only decision we could make.”