Welbeck traded Old Trafford for the Emirates Stadium on the final day of the transfer window on Monday in a move reported to be worth 16 million pounds.
The England striker had previously spent his entire career at United, save for loan spells with Preston North End and Sunderland, but was rarely able to establish himself as a regular.
Welbeck made just 24 starts in all competitions last season, scoring 10 goals, but is now hoping to make the most of his undoubted potential in north London.
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"It's exciting times for me," Welbeck told Arsenal's official website. "It's great to be a part of this club and it's a team that I've always watched in the Premier League. I've envisaged myself playing in this team before. For it to finally happen is very exciting. 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.
"I've been at United ever since I was a little boy and I had a great time there. I just think this is the next stage of my development - to come to Arsenal, to really show people what I can do as a footballer and hopefully get to the place where I want to be."
Wenger's men ended a nine-year trophy drought by winning the FA Cup in May before claiming the Community Shield last month.
Welbeck, who has one Premier League title to his name, is hopeful he can add to that tally with his new employer.
He added: "They've [Arsenal] 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.
"Things move on in life and I think you've really got to make the right decision in the situation that you're in at that moment in time. I really do believe that coming to Arsenal is the right decision."