The Red Devils brought an end to their run of straight defeats since the start of 2013 with an assertive performance against Michael Laudrup's men.
The focus before the game had almost exclusively centered on the United boss following the club's wretched run of recent form that had seen the club lose its first three fixtures of the New Year.
United, though, at last brought an end to the run against the side that only a week beforehand had eliminated it from the FA Cup.
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Moyes's side responded well to a miserable week and is now four points behind fourth-placed Everton, while Swansea is down in 13th place and just three points above the relegation zone after now having failed to win in five top-flight games.
Chris Smalling replaced Jonny Evans (knock) in United's starting lineup following the defeat at Sunderland, while Darren Fletcher and Shinji Kagawa were preferred to Ryan Giggs and Tom Cleverley.
Captain Ashley Williams returned to the Swansea side as Michael Laudrup made four changes to the team that won at Old Trafford last weekend.
United made a positive start and Nemanja Vidic fired over the crossbar from close range six minutes in after Swansea had failed to deal with Adnan Januzaj's dangerous free kick.
Teenager Januzaj went close to an opening goal four minutes later when his measured, left-footed free kick rattled the crossbar and Swansea suffered a blow six minutes later, as Jose Canas limped off to be replaced by Chico Flores.
Patrice Evra called Gerhard Tremmel into action after 27 minutes, but the defender's shot was comfortably dealt with by the Swansea goalkeeper.
Swansea was enjoying plenty of possession, but Welbeck should have put United in front 10 minutes before halftime when he fired wide of the far post from eight yards out after marauding right back Rafael had picked him out.
The tricky Brazilian then produced some neat footwork before swinging in a cross that Kagawa headed back across goal and Tremmel could only palm the ball into the path of Valencia, who was left with a simple finish from close range to ease the tension at Old Trafford.
Kagawa surged forward and fired over the crossbar after a slick break involving the impressive Januzaj and Welbeck as United raised the tempo.
The home club doubled the lead after 58 minutes, Januzaj involved once again as his cross was headed away and Welbeck was on hand to cutely guide the ball into the net after Evra had knocked it back into the area.
Smalling should have added a third goal after 65 minutes, but the defender inexplicably fired over the bar from point-blank range when it looked easier to score after he was picked out by Kagawa.
Kagawa then wasted a golden opportunity of his own when he was played in by Rafael, but his casual finish was cleared off the line.