It was a horror moment for United goalkeeper David de Gea as he allowed Van Aanholt's powerful shot to slip through his grasp and creep into the net to secure Palace's first league win over United for 23 matches and their first at Old Trafford since December 1989.
The Dutchman's goal came after it looked like Daniel James had rescued a point with an 89th-minute equaliser and saved the blushes of Marcus Rashford, who missed a penalty as United suffered more problems from 12 yards.
The last few days had been dominated by the fallout of Paul Pogba's failure against Wolves on Monday and whether he should have stepped up instead of Rashford, who had scored from the spot against Chelsea on the opening day.