Several Arsenal fans have voiced doubts about the signing on social media, and Welbeck was one of the players singled out for criticism after England's dismal World Cup campaign in Brazil, but the 23-year-old was defiant.
"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," he told the Arsenal website (www.arsenal.com).
"I've been at United ever since I was a little boy and I had a great time there. Coming to Arsenal is the right decision. I can't be any happier at the moment," he said.
Welbeck became surplus to requirements at Old Trafford as new manager Louis van Gaal brought in Colombian striker Radamel Falcao on loan from Monaco but the England international said Arsenal would suit him.
"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," he said.
"It's exciting times for me. 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."
Welbeck's move was given enthusiastic backing by former United team mate Rio Ferdinand.
"Welbeck is a top signing for Arsenal. Good business done by them. If he gets that trust to be no.1 striker at AFC he will flourish & explode," the 35-year-old defender, who signed for Premier League club Queen's Park Rangers in July after 12 years at Old Trafford, said on Twitter.
Currently on international duty with England, who face Norway in a friendly on Wednesday, Welbeck will join up with his new team mates to prepare for the league match at home to Manchester City on Sept. 13.