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 points 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.
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