Following Welbeck's introduction at halftime, United looked much more balance going forwards and got the crucial breakthrough on 57 minutes when a clearance from Ryan Bennett bounced back off the England striker and 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.
"We would have started him, but he has played a lot of games, but we are trying to make sure that we give the squad the right opportunities to play and make sure we give them chances to show what they can go.
"Danny has been playing very well and came on to make the difference today."
Moyes added: "If you look at Danny now, he is a big part of Manchester United.
"He is a really good player, what he needs to become now is a really good finisher. Today he rounded the goalkeeper (for his goal) and was unlucky with one which came to the far post and was not able to get a finish away there.
"Danny had played really well at Hull the other day, but he just looked a bit fatigued and so we decided not to start him today."
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