United remained in seventh place with 37 points from 21 games, nine points behind leaders Chelsea, but victory at least improved the mood at Old Trafford, even if the performance was not memorable.
"In the end we got the result we deserved," Moyes, who was charged with misconduct by the FA on Friday for comments he made about officials after Tuesday's League Cup semi-final, first leg defeat at Sunderland, told Sky Sports.
"At halftime I said we had to work a bit harder and make more chances and they (the players) went about their jobs very well."
Michael Laudrup's injury-hit Swansea dominated possession with their patient passing style but apart from a Jonjo Shelvey shot easily held by David de Gea they created few chances.
United also struggled, however, with the game looking eerily similar to last Sunday's FA Cup third round tie at Old Trafford that Swansea won 2-1 with a late Wilfried Bony goal.
There was to be no repeat on Saturday though as United, with teenager Adnan Januzaj switched from a central role to the wing at halftime, took the lead two minutes after the interval.
Valencia was on hand to tuck away a rebound after Shinji Kagawa's header was parried by Swansea keeper Gerhard Tremmel.
"Not many have dropped for us in the box lately and sometimes you need that bit of luck," Moyes said.
Welbeck glanced in Patrice Evra's shot on the hour to give United daylight and they should have added more goals late on.
If Moyes was a relieved man after another week of intense scrutiny he was not showing it, however.
Asked how big the victory was, the Scot simply said: "Same size as any other one. All you can get is three points."
United visit Chelsea next week when they hope to have Rooney available after a "warm weather" break to heal a groin problem.
(Writing by Martyn Herman; Editing by Ken Ferris)
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