Despite getting run ragged for most of the afternoon, and seeing midfield duo Michael Carrick and Tom Cleverley disappear without trace, United nicked the points thanks to the increasingly talismanic Welbeck's 10th goal for club and country this season.
This was United's sixth straight victory and moves them back within striking distance of the top four.
The goal came 12 minutes into the second half when Javier Hernandez blocked a Norwich clearance and the ball ricocheted through to his striker partner who rounded the goalkeeper and shot into the empty net.
For the second time in 48 hours David Moyes's side have earned maximum points despite performing poorly - on Boxing Day they came from two goals down to beat Hull.
While Moyes will be delighted to see his side regaining their ability to play badly and win, the mediocrity of their displays means they remain distant outsiders for the Premier League title.
17' - CHANCE: Snodgrass feeds the ball into Russel Martin who let's fly with a shot that is saved by David De Gea. Wes Hoolahan fails to convert the rebound.
21' - SAVE: De Gea makes another save as Bradley Johnson finds Hoolahan in the middle then Snodgrass has a shot with his left foot but sends it well wide.
57' - GOAL (Norwich 0-1 United): Ryan Bennett clears the ball on to the leg of Welbeck, the balls ricochets off Hernandez and then back into the path of Welbeck who rounds Ruddy and slots home.
71' PENALTY? Mild appeal by Norwich as Hoolahan was muscled out of the area but the referee isn't interested.
How does Chris Hughton break the deadlock? Norwich looked dominant for large parts of the match but, as on Boxing day, will rue the missed chances. A concern for Hughton will be that leading scorer Gary Hooper had to come off with a hamstring problem but there are signs that young Josh Murphy has something about him.
Norwich had 15 shots (four on target) compared to Manchester United's 7 shots (three on target); they won 8 corners to United's three.
Norwich are now without a win in four league games and whilst they remain 14th they are just three points off the drop zone.
This was Welbeck's 17th league goal for United.
MAN OF THE MATCH
Danny Welbeck (Manchester United) - Moyes's half-time sub made all the difference to this match, making United look much more balanced. They had struggled in the first half, acutely missing the services of Wayne Rooney, but Welbeck's introduction brought Hernandez much more into the game as they worked in partnership.
Norwich: Ruddy 6, Martin 6, R Bennett 6, Bassong 6, Olsson 6, Fer 6, Johnson 6, Snodgrass 7, Hoolahan 7, Redmond 6, Hooper 6; Subs: Van Wolfswinkel 6, Elmander NA, Murphy 6
Manchester United: De Gea 7, Smalling 7, Vidic 6, Evans 6 , Evra 5, Young 6, Cleverley 6, Carrick 6, Giggs 6 , Kagawa 6, Hernandez 7; Subs: Welbeck 7, Fletcher NA, Januzaj 6
MATCH IN A TWEET