The Reds started brightly and could have taken an early lead through Nemanja Vidic, who volleyed over from six yards, before the impressive Adnan Januzaj struck the crossbar with a delicious free-kick moments later.
With just over half an hour on the clock, a lovely passing move on the edge of Swansea's area led to Darren Fletcher slicing high and wide, before Welbeck dragged another effort wide of goal, missing the first half’s best chance following some neat approach play from Valencia and Rafael.
David Moyes didn’t make any changes at the break, though his side returned for the second half with renewed urgency and immediately scored via Valencia, who slotted in at the far post after Shinji Kagawa’s header was parried into his path by Gerhard Tremmel. Welbeck doubled the lead on 59 minutes, much to the delight of a relieved Old Trafford crowd, by expertly clipping Patrice Evra’s speculative shot into the net for his ninth goal of the season.
Chris Smalling, playing at centre-back, missed an excellent opportunity to score a third when his close-range volley sailed over the bar, while Kagawa should have opened his account for the season on 77 minutes when his delicate effort was cleared off the line after a superb counter-attack involving Januzaj and Rafael.
It may not have been a classic, but a first victory of 2014 was all that mattered for United.