United appeared to be clinging on for a point as Villa created a succession of chances after Douglas Luiz had cancelled out Rasmus Hojlund's first-half strike.
United manager Erik ten Hag punched the air in delight as his side moved to within five points of their hosts, who remain fifth, and six adrift of Tottenham.
It was United's fourth successive victory in all competitions - their best run of the season.
The result was harsh on Villa, though. They were thwarted on numerous occasions by United keeper Andre Onana, and despite the urgings of the home support were unable to draw level a second time.
More to follow.