Beaten by Everton and Newcastle United most recently, United were at risk of losing three successive games for the first time since 2001, but were dominant at Villa Park.
A quick-fire brace from Danny Welbeck put United in control early and they added a further goal through Tom Cleverley to close to within 10 points of leaders Arsenal.
Victory left the champions six points below the Champions League positions in eighth place, but this was a timely win for manager David Moyes after a desperate run of four games without victory.
Marc Albrighton worked United goalkeeper David de Gea twice in the early stages, but - deputising for injured Robin van Persie - England forward Welbeck broke the deadlock in the 15th minute when he slammed home after Adnan Januzaj had headed against the post from a Rafael da Silva cross.
Villa were unable to contain United's right-sided players and, three minutes later, the visitors scored another goal, with Antonio Valencia crossing for the sliding Welbeck to score.
Wayne Rooney, playing his 500th club game, and Welbeck both went close to extending United's lead before Cleverley added a third in the 52nd minute with a first-time finish from Rooney's pass.
It was the England midfielder's first goal for a year and United celebrated another milestone later in the game when Darren Fletcher came on to make his first appearance after a year out with a chronic bowel condition.
Villa, who have not beaten United at home since 1995, remain 11th on the table.
After leaders Arsenal's 6-3 capitulation at Manchester City on Saturday, Liverpool can return to second place and close to within two points of the summit if they win at Tottenham Hotspur later on Sunday.
In the day's other game, Swansea City drew 1-1 at Norwich City in a game overshadowed by an injury to visiting goalscorer Nathan Dyer.
He broke the deadlock with a 12th-minute lob, but had to be carried off on a stretcher and taken to hospital with a suspected fractured ankle after a duel with Sebastien Bassong.
Gary Hooper levelled for Norwich in first-half stoppage time, finding the net from 25 metres with a dipping volley.