United rode their luck at times and Welbeck’s goal had an element of good fortune about it, but the champions held on for their sixth successive victory in all competitions.
The reigning Premier League champions are up to sixth, two points short of a top-four place, although four of the five clubs above them now boast a game in hand.
United boss David Moyes made several changes to the side that fought back to win 3-2 at Hull on Boxing Day.
Wayne Rooney, who scored his 150th United goal in that game, did not travel because of a groin strain and Antonio Valencia missed out through suspension as Javier Hernandez, Shinji Kagawa and Ashley Young started.
Norwich, who beat United 1-0 at Carrow Road last season, also changed their starting line-up following a 2-1 home loss to Fulham, with Wes Hoolahan, Russell Martin and Ryan Bennett coming in.
The struggling Canaries looked nervous and were unable to hold on to possession in the early stages.
United, with Ryan Giggs controlling midfield, tested the hosts on both flanks and Japan midfielder Kagawa saw a shot from a tight angle blocked by Canaries goalkeeper John Ruddy.
The home side calmed their nerves eventually and almost took the lead in the 20th minute when Russell Martin, having played a one-two with Robert Snodgrass, forced David de Gea into a parrying save.
Chris Smalling made a mess of clearing the rebound at the back post but was able to hook the ball away before Wes Hoolahan could pounce.
United had to soak up some intense pressure after that, which saw Gary Hooper’s fierce drive saved by de Gea and Snodgrass twice fail to hit the target.
Referee Phil Dowd then booked three players in the space of a few minutes – Giggs and Hernandez plus Norwich midfielder Leroy Fer – but did not award Norwich a penalty when Tom Cleverley brought Bradley Johnson down while trying to clear.
Norwich continued to dominate and de Gea had to get behind a header from a corner from Sebastien Bassong before Snodgrass shanked the last chance of the half wide.
Welbeck replaced Giggs at the break as United switched to a 4-4-2 formation and that proved a key moment.
Young perked up and a run ended with a shot that only just cleared Ruddy’s crossbar.
Nathan Redmond was inches away from opening the scoring at the other end but it was United who found the net in the 57th minute through Welbeck.
The substitute blocked Bennett’s clearance, with the ball then bouncing off Hernandez and allowing Welbeck to round Ruddy and score.
United now looked the more likely to add to the score-line and Ruddy had trouble dealing with a Hernandez long-range strike.
Norwich boss Chris Hughton made his last roll of the dice, with Ricky van Wolfswinkel replacing Hooper to derisive jeers from the unhappy home fans.
Netherlands international van Wolfswinkel was unable to make much of an impact as United saw out the closing minutes with little alarm.