With Wayne Rooney not travelling to Norfolk because of a groin problem and Robin van Persie still not match fit, United lacked a cutting edge in attack during a first half in which the Canaries were well on top.
However, following Welbeck's introduction at the restart, United looked much more balance going forwards and got the crucial breakthrough on 57 minutes when a clearance from Ryan Bennett bounced back off the England striker and into the penalty area, where he rounded John Ruddy to slot home.
It was a sixth straight victory for David Moyes' men who are now back within striking distance of the Barclays Premier League top four heading into the new year.
"Obviously his goal made the difference, but Danny's performance changed the game," said Moyes, who expects Rooney to be back for the clash against Tottenham on New Year's Day.
"We would have started him, but he has played a lot of games, but we are trying to make sure that we give the squad the right opportunities to play and make sure we give them chances to show what they can go.
"Danny has been playing very well and came on to make the difference today."
It was a fourth goal in five appearances for Welbeck, who has often had to play understudy to both Rooney and Van Persie this season.
Moyes, though, is in no doubt of the importance of the 23-year-old to the Red Devils' attacking options.
"If you look at Danny now, he is a big part of Manchester United," said the United manager.
"He is a really good player, what he needs to become now is a really good finisher. Today he rounded the goalkeeper (for his goal) and was unlucky with one which came to the far post and was not able to get a finish away there.
"Danny had played really well at Hull the other day, but he just looked a bit fatigued and so we decided not to start him today."
Moyes hopes Rooney's lay-off will be a shortlived.
"I would expect Rooney to be back for Wednesday, but he has got an abductor injury in his groin which kept him out of the Stoke game," he said.
"Ideally we would like to give it a bit longer than three of four days between each game, but we are not getting much time at this point of the season.
"Robin has started some light training and he might not be too far away as well.
"We have not had them as a partnership very often, and we are looking forward to having them together."
After a slow start to his tenure as United boss following Sir Alex Ferguson's retirement, Moyes' new-look side are slowly finding their rhythm.
"Manchester United are now winning games, they did that last season, and they won the league not always being at their best, today they found a way to win the game," he said.
"To win two away from home (over Christmas) takes a lot of doing we have had a few injuries and a lot of our key players out.
"I am delighted with how we are doing and want to keep it going, but I don't think it is anything to shout about."
Moyes added he was not planning a major spending spree when the transfer window opens again.
He said: "We have a lot of midfield players out injured just now, but we won't be doing a lot of business in January because a lot of the players will not be available, so I don't expect it to be a busy January really."
Norwich could feel somewhat aggrieved not to have taken something from a match for which they were on top for long spells.
Having lost at home to Fulham on Boxing Day, the pressure now increases on Chris Hughton's men for a positive result at Crystal Palace on New Year's Day.
He said: "It is certainly a game we didn't deserve to lose.
"We were excellent in the first half, but when you are having your best period, you have got to capitalise on that and we didn't.
"It was a fortunate goal for them - a double deflection - and when that falls to somebody of Welbeck's quality, then you are almost expecting him to score.
"That is the only difference between the two teams, and over the 90 minutes we were the better side."