The injury-hit Reds were rarely at their best during a largely dour affair, but improved markedly on a poor first half display after the break, and were rarely troubled after Welbeck had put David Moyes' side into the lead shortly before the hour.
The manager's preparations were dealt a blow by the absence of Wayne Rooney, who remained in Manchester with a minor groin injury, while Antonio Valencia was also absent through suspension. While Michael Carrick's first start since early November provided a timely boost, United were disjointed for much of the first period.
The Reds' two best chances of the first half were separated by 40 minutes. Between those openings - a Shinji Kagawa cross-shot beaten away by John Ruddy and a tame near-post Chicharito effort - Norwich overcame a timid opening to steadily take control of the game. David De Gea needed to thwart Russell Martin, Gary Hooper and Sebastian Bassong, while Jonny Evans bravely blocked a Wes Hoolahan shot and Robert Snodgrass repeatedly chanced his arm from distance without success.
The Canaries had blunted United's attacking play by regrouping quickly in-between their own periods of dominance, but the Reds grasped control of the game with the half-time introduction of Welbeck. The substitute added much-needed pace and penetration to United's approach, and he quickly made a telling difference by charging down Ryan Bennett's clearance and racing onto the loose ball, via a Chicharito touch, before rounding Ruddy and sliding home a finish to settle the game in United's favour and ensure a six-game winning streak to end 2013.