Casemiro broke the deadlock at Old Trafford with his goal after 53 minutes.
Ross Barkley drew the scores level with his strike after 64 minutes, but Matheus Cunha restored the home side's lead before Benjamin Sesko secured the result when he netted with nine minutes of the match remaining.
The result saw Michael Carrick's team move above Villa into third place on the standings with 54 points from 30 games, and the Villans in fourth with 51 points.
Carrick, meanwhile, became only the fourth manager in Premier League history to win his first six home games in a row - and it was a welcome result after losing 2-1 to Newcastle on the road on Wednesday.
The home team bossed possession from the opening whistle but without really threatening their opponents.
The first real chance fell to Amad Diallo on 17 minutes but he was unable to get his head to Matheus Cunha's superb cross from the left.
Amad though was on target from a similar position moments later with a strong header that forced a strong one-handed save from Villa goalkeeper Emiliano Martinez.
Diogo Dalot blasted over from close-range after being set up by Bruno Fernandes while only Leny Yoro stopped United from heading down the tunnel at half-time a goal down when he pulled off a goal-saving tackle to stop Ollie Watkins from firing home.
United started the second half on the front foot and after a few half-chances, they ultimately took the lead within 10 minutes of the restart.
Casemiro reacted quickest to get his head to Bruno Fernandes' near post corner delivery and leave Martinez with no chance.
Villa responded with Amadou Onana testing Senne Lammens with a fierce drive that he kept out well.
But the United stopper he could do nothing to stop Barkley from firing into the back of the net just moments later.
Despite the set-back, United continued to press and well rewarded with a second goal when Cunha made no mistake with a strike into the corner after being played in by Fernandes.
Substitute Benjamin Sesko then came off the bench and after some luck and a few deflections, was able to see the ball over the line.
The result put United just six points behind second-placed Manchester City and 16 points adrift of league leaders Arsenal.
Max Dowman made Premier League history as the 16-year-old helped Arsenal secure a dramatic 2-0 win over Everton at the Emirates Stadium, with manager Mikel Arteta revealing a pre-match "gut feeling" led him to trust the teenager.
Chelsea head coach Liam Rosenior has criticised referee Paul Tierney after a bizarre pre-match moment saw the official standing in the middle of the Blues' team huddle before their clash with Newcastle United.
Manchester City failed to close the gap with top of the table Arsenal after a 1-1 draw in their Premier League match against West Ham United the London Stadium on Saturday.
Arsenal struck two late goals to down Everton 2-0 and keep up their Premier League title-winning pace, as Newcastle backed up their victory over Manchester United by edging Chelsea 1-0 at Stamford Bridge.
