Trailing 2-1 from the 34th minute, Palace battered at the door after the break until the visitors’ resistance was finally broken at Selhurst Park.
With 15 minutes left, Wilfried Zaha produced a driving run before cutting inside and firing a fierce shot that deflected into the net off Villa defender Ahmed Elmohamady.
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Then, with six minutes to go, Mitchell found himself in the perfect spot after a fluffed Eberechi Eze cross to bundle the ball home with his shoulder.
Villa remain 11th, four points behind 10th-placed Leeds with two games to play.
On a more positive note for Villa and potentially England, Jack Grealish made a second successive appearance from the bench following injury to provide further evidence of his improving fitness before this summer’s European Championships.