The deal was sealed in the early hours of Tuesday morning after the Gunners were given a two-hour extension to the transfer deadline to complete all the formalities.
Welbeck, 23, had been the subject of interest from a host of Premier League clubs after being told he was no longer wanted under new United boss Louis van Gaal.
Tottenham were seriously interested in signing Welbeck on loan, while Newcastle, Sunderland and Hull City also made enquiries, but Arsenal moved for the forward late on Deadline Day.
Sky sources reported earlier on Monday that the player had left training with England in Hertfordshire to undergo talks with the Gunners and he has now signed a long-term contract with the north London club.
Welbeck came through the youth ranks at United and rose to prominence on a successful season-long loan at Sunderland during the 2010-11 season.
He has scored 20 Premier League goals in 90 appearances and has won 26 caps for England since his 2011 debut, scoring eight.
However, his only start this season came in the 4-0 League Cup thrashing at MK Dons last month.