The England striker replaced Ryan Giggs at the interval at Carrow Road after a poor first half from the reigning champions, who were without Wayne Rooney due to a recurrence of his groin injury.
Welbeck then took the ball round John Ruddy to score the only goal of the game on 57 minutes, as United won their sixth straight game to move up to sixth in the table. Norwich remain 14th, three points above the drop zone at the midway point of the season.
Norwich boss Chris Hughton made three changes to the Canaries side that started the 2-1 home defeat by Fulham on Boxing Day. In came Russell Martin, Ryan Bennett and Wes Hoolahan, with the banned Michael Turner making way, and Steven Whittaker and Johan Elmander dropping to the bench.
David Moyes made five changes to the United team that started the 3-2 win at Hull on Thursday. Nemanja Vidic, Javier Hernandez, Michael Carrick and Shinji Kagawa - who were all subs at the KC Stadium - came into the starting line-up, along with Giggs. Welbeck and Darren Fletcher dropped to the bench, Antonio Valencia was banned, while Rooney - who should return soon - and Rafael (also groin) were ruled out.
After Martin almost turned the ball into his own net, Norwich had the better of the first half.
On 17 minutes, Martin was fed into the right side of the area by Robert Snodgrass and let fly with a shot that was saved by David De Gea before Chris Smalling bundled the ball out for a corner, Hoolahan having failed to turn it home.
Four minutes later, De Gea had to make another save, this time to deny Gary Hooper from inside the area. Leroy Fer fed the ball right to Martin, and then Snodgrass crossed in low to set up the chance.
Giggs, Fer and Hernandez were all booked midway through the half as referee Phil Dowd asserted his authority.
De Gea saved a Seb Bassong header from a corner on 36 minutes, and then United managed their first shots on goal just before the break. Carrick and Smalling were off target, while Hernandez's effort from the left side of the area was saved by Ruddy.
For the second half, Welbeck was introduced at the expense of Giggs and Ashley Young attempted to get United going - but his crossing from the right and his shooting were something of a mixed bag.
At the other end, winger Nathan Redmond's low effort curled through the United six-yard box and just wide of the near post.
However, United did make the breakthrough on 57 minutes after a slice of good fortune. Bennett cleared the ball into the leg of Welbeck, the ball ricocheted off Hernandez and back into the path of his strike partner. Welbeck then took the ball around Ruddy expertly to open the scoring with his fourth goal in five games.
Ricky van Wolfswinkel replaced hamstring injury victim Hooper on 69 minutes, while Adnan Januzaj came on for Kagawa.
But other than more fussy refereeing from Dowd, there was little incident in the closing stages.
Eighteen-year-old winger Josh Murphy was introduced to try to inject some fresh impetus into the Norwich attack, with Snodgrass substituted. Fletcher came on for Hernandez for the final few minutes as United successfully saw out the game.
Norwich move on to an important game at Crystal Palace on New Year's Day, while United host Tottenham in an evening kick-off on Wednesday.