"Obviously his goal made the difference but I thought Danny's performance changed the game," Moyes told a news conference.
"We've been trying to make sure we give the squad the right opportunities to play and give them opportunities to show what they can do. Danny has been playing really well and he came on and made the difference today."
England international Welbeck, 23, has often been criticised for failing to score enough goals.
"He's a big part of Manchester United," Moyes said. "I said the other day Danny is a really good player and what he needs to become is a really good finisher and today he did great, rounded the goalkeeper."
United completed their sixth successive win in all competitions to climb to sixth in the table after a poor start to the season.
"That's six points from two games in three days, both away from home," Moyes said.
"I thought Norwich did terrific today but overall we kept going, we kept at it and we got our rewards in the end. Today we found a way of winning the game despite a lot of injuries and a lot of players out."
Moyes is confident Rooney will be fit for the midweek home game against Tottenham Hotspur.
"I would expect Rooney to be back on Wednesday," he said. "He has got an adductor injury in his groin.
"Ideally we'd like to give it a bit longer, three or four days between each game doesn't give you much time at this stage of the season."
Rooney's strike partner Robin van Persie is also close to a return after six weeks out with a thigh injury.
"Robin has started some light training and he might not be too far away as well," Moyes said.
"We've not had them as a partnership very often and we're looking forward to getting them back together."
Norwich's defeat left them in 14th spot.