The second-half substitute's sixth Premier League goal of the season saw David Moyes' side secure a fourth consecutive top-flight win to close in on the top-four places.
Having replaced Ryan Giggs at halftime, the 23-year-old England striker made an almost immediate impact, netting his fifth goal of the campaign in the Premier League to move United into sixth place in the table.
Norwich had dominated the opening half but wasted a host of chances to take the lead, and the team's profligacy eventually came back to haunt it as United clicked into gear after the break.
Welbeck had a bit of luck in the build-up to the goal, as the ball bounced off Javier Hernandez into his path, but he showed great composure to open the scoring from there.
The introduction of Ricky van Wolfswinkel failed to have a similar effect for the home side, which slipped to a fourth home league defeat of the season.
Norwich manager Chris Hughton made three changes to the side beaten by Fulham with Russell Martin, Ryan Bennett and Wes Hoolahan coming in for Steven Whittaker, Johan Elmander and the suspended Michael Turner.
With Wayne Rooney ruled out with a groin strain - he is expected to be back to face Tottenham on New Year's Day - United boss David Moyes recalled Shinji Kagawa, with the Japan international returning as one of five changes made by the Scot as the team went in search of a fourth consecutive win in the league.
After a cagey start, the first chance of the game came in the 17th minute as Martin was released by Robert Snodgrass inside the penalty area and his shot was palmed into the path of Hoolahan by David De Gea, but the Republic of Ireland international failed to take advantage of an open goal.
Five minutes later, De Gea was called into action again as the Martin-Snodgrass combination again worked well to release the winger behind the defense. He rolled the ball back to Gary Hooper, whose shot on the turn was parried away.
Norwich continued to apply the pressure with Sebastien Bassong and Hoolahan both missing further chances, while only Hernandez could force a save from John Ruddy for United.
Ashley Young and Nathan Redmond both went close in the opening stages of the second half, but it was Welbeck who finally broke the deadlock in the 57th minute.
The striker charged down a Bennett clearance, the ball ricocheted off Hernandez back to the substitute and he rounded Ruddy before sliding the ball into the back of the net.
Hughton's attempt to lift his side late on with the introduction of Van Wolfswinkel and Elmander failed to have the desired effect, allowing United to secure a sixth away win in the league this season.