With England forward Wayne Rooney not travelling to Norfolk because of a groin problem and Robin van Persie still not match fit, United lacked a cutting edge in attack during a first-half in which the Canaries were well on top.
However, following Welbeck's introduction at the restart, United looked much more balanced going forwards and got the crucial breakthrough on 57 minutes when a clearance from Ryan Bennett bounced back off the England striker and on into the penalty area, where he rounded John Ruddy to slot home.
It was a sixth straight victory for David Moyes' men who are now back within striking distance of the Barclays Premier League top four heading into the new year.
Despite all their changes from the 3-2 win at Hull on Boxing Day, with captain Nemanja Vidic back in defence and midfielder Michael Carrick starting his first match since mid-November, United had started brightly.
On six minutes, Ashley Young got away down the right and his cross found its way through to Shinji Kagawa at the far post. The Japan forward cut the ball back through the six-yard box, where Norwich goalkeeper Ruddy palmed it clear.
Norwich - with Irish playmaker Wes Hoolahan recalled to a five-man midfield in support of Gary Hooper - offered little attacking threat in the first quarter-of-an-hour as too many passes went astray.
Suddenly, though, the Canaries carved United open down the right as Robert Snodgrass played in Russell Martin.
The full-back's angled drive was palmed away by David de Gea and the loose ball was just out of reach for Hoolahan as it was bundled behind for a corner.
Hooper's snapshot from six yards was then straight at De Gea as Norwich continued to press, with Snodgrass curling wide from the edge of the United area.
Norwich were enjoying the better of possession as the half-hour passed.
Captain Sebastien Bassong also found the United goalkeeper with a close-range header before Hoolahan's shot after a weaving run into the penalty area was blocked by Jonny Evans.
United had looked lacklustre in attack, but made a positive change for the second half as Ryan Giggs was replaced by Welbeck.
Young drove forwards and flashed a 20-yard shot just over Ruddy's crossbar.
At the other end, winger Nathan Redmond's low effort curled through the United six-yard box and just wide of the near post.
United took the lead on 57 minutes.
Centre-half Bennett - in for the suspended Michael Turner - crashed his clearance against Welbeck, which cannoned back off the United forward and on to Hernandez.
The ball then bounced through towards goal, where Welbeck sprinted into the area to round Ruddy and slip the ball into the empty net for his fourth goal in five appearances.
Norwich's energy was clearly drained, having seen all of their earlier hard work undone in a moment of poor fortune.
United, though, were in cruise control as they looked to close in on a sixth straight win, with Hernandez's 25-yard effort testing Ruddy.
With 22 minutes left, both teams made changes as Norwich sent on club-record signing Ricky van Wolfswinkel to replace Hooper and Adnan Januzaj came on for Kagawa.
Eighteen-year-old winger Josh Murphy was introduced to try to inject some fresh impetus into the Norwich attack for the final 11 minutes, with Snodgrass substituted.
The late rally never materialised, though, as United closed out what was in the end a comfortable victory to leave Norwich looking over their shoulders at the battle for survival once again.