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 champions were not at their best in the first half, but Valencia opened the scoring just after half-time with his first goal at home in the league since December 2011 and Welbeck scored his sixth in as many league games as Moyes' men secured their first win of the year.
Chris Smalling replaced Jonny Evans (knock) in United's starting line-up following the defeat at Sunderland, while Darren Fletcher and Shinji Kagawa were preferred to Ryan Giggs and Tom Cleverley in midfield.
|VIEW FROM OLD TRAFFORD
|By Peter Staunton
It is a happy end to the week for Manchester United who have bounced back from three consecutive defeats to beat Swansea 2-0 at Old Trafford. Goals early in the second half from Antonio Valencia and Danny Welbeck ease the pressure on David Moyes. Swansea had some chances of their own but failed to test David de Gea as the hosts stood resolute.
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.
The visitors were enjoying plenty of possession, but Welbeck should have put United in front 10 minutes before half-time 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.
And they doubled their 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 the ball 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, before Kagawa also wasted a golden opportunity as United had to settle for just the two goals, but a precious three points.