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Welbeck, 23, will wear the No.23 for the Gunners and could make his debut against Manchester City at the Emirates Stadium a week on Saturday.
"I've envisaged myself playing in this team before. For it to finally happen is very exciting," Welbeclk told Arsenal's website.
"I believe the style of play the manager's got and the boys play and with the magnificent players in midfield slotting balls through, I can run on to the end of those balls and slot them away.
"Arsenal have done well in recent years and last year we got the FA Cup and I feel like it's time where we could really push on and try and challenge for the Premier League title. I've got one medal and it's something that I really want to add to. "
The England international had been agitating for a move away from United since April after becoming frustrated with his lack of game-time under David Moyes, despite Robin van Persie's injuries.
Although the majority of United supporters wanted Welbeck, a local lad who was spotted by the club when he was six, to stay, Radamel Falcao's arrival escalated his departure.
Welbeck scored 29 goals in 152 appearances for United, whom he made his debut for six years ago in a League Cup win over Middlesbrough.
On the two occasions he faced Arsenal Welbeck scored. He opened the scoring in the memorable 8-2 rout in 2011 and later that season struck the winner at the Emirates.Suggest a correction