Goal understands that a deal was struck late on Monday night to bring the England international to the Emirates Stadium on a four-year contract, with an option for a fifth.
Welbeck is set to earn around £100,000 per week at Arsenal, who have successfully secured a deal under the noses of north London rivals Tottenham.
Arsene Wenger moved for the 23-year-old after failing in efforts to sign Loic Remy and Radamel Falcao, the latter of whom is set to join United on a season-long loan deal.
The Gunners boss had previously denied that he would move for either Welbeck or Falcao but has since decided to snap up cover up front, with Olivier Giroud set to miss around four months due to a broken tibia.
Spurs had been in advanced talks to sign Welbeck on loan but Goal understands that Arsenal spent much of deadline day persuading United to sanction a permanent exit.
The forward had grown frustrated at a lack of first-team opportunities and had requested showdown talks with new manager Louis van Gaal earlier this summer to resolve the issue.