Back-to-back defeats against Everton and Newcastle United had left Moyes' men ninth in the table, but they got back on track at Villa Park by securing their first win in five top-flight games.
Welbeck had not scored for over two months and was without a league goal since August, yet the England striker struck twice in the space of three first-half minutes on Sunday.
Tom Cleverley then scored his first Premier League goal for exactly a year early in the second half to end any hopes of a Villa comeback and Darren Fletcher made his first appearance since Boxing Day after recovering from a bowel problem on a morale-boosting afternoon for United.
Last year's champions have now won their last six games against Villa and the latest victory ensures they leapfrog Southampton to move into eighth, 10 points behind leaders Arsenal. Villa have already suffered six home defeats this season.
Fletcher was among the United substitutes and Moyes made four changes to the side that beat Shakhtar Donetsk in midweek as Antonio Valencia, Patrice Evra, Cleverley and Welbeck started. Marc Albrighton made his first Villa start of the season as Paul Lambert also made four alterations.
United were in front after only 15 minutes when Rafael da Silva got to the byline and whipped in an inviting cross. Adnan Januzaj headed against the post, but Welbeck slotted in the rebound.
Villa were stunned three minutes later when United doubled their lead. This time Antonio Valencia was the architect as he burst down the right before delivering the ball into the path of the onrushing Welbeck, who slid in to finish clinically from close range.
Valencia played a one-two with Welbeck but was indecisive and failed to get his shot away after 27 minutes.
Soon after, the winger set up Rooney and the England striker flashed a shot wide of the far post.
Welbeck had a great chance to complete his hat-trick after only 33 minutes when he raced clear but dragged his shot wide of the far post.
United were out of sight seven minutes after the break as Rooney played in Cleverley and the midfielder tucked the ball inside the near post with a measured right-foot finish.
Cleverley then showed his value at the other end as he produced a timely block to thwart Andreas Weimann.
Fletcher was given a long-awaited run-out with just over 20 minutes remaining and Welbeck had another chance after 72 minutes, but his lob floated over the crossbar.
David De Gea palmed away Karim El Ahmadi's curling free-kick 14 minutes from time and Cleverley stung Brad Guzan's palms as United saw out a comfortable victory.
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