Daniel James thought he'd restored parity when he coolly curled the ball into the top corner with one minute of normal time remaining, but Van Aanholt then sensationally added to Jordan Ayew's first-half effort for a famous 2-1 win.
There was more penalty drama for United, too. This time, Marcus Rashford stepped up from 12 yards, but his 70th-minute attempt hit the inside of the post and somehow bounced out for a goal kick. Rashford was handed spot-kick duties five days after the overblown scrutiny which followed Paul Pogba's failed attempt at Wolverhampton Wanderers.
The result marked the first time the Eagles took three points away at Manchester United since December 1989, when Ole Gunnar Solskjaer's right-hand man, Mike Phelan, was playing in midfield and none of the Red Devils' starting XI were born. Crystal Palace hadn't even beaten Manchester United at Selhurst Park in the Premier League era.
"There is a first time for everything. I think we played really well. I think we deserved the three points," Van Aanholt said after the match, according to BBC Sport.
The Eagles aimed five shots at De Gea's mesh, while only three of Manchester United's 22 shots hit the target.
Solskjaer's side unsurprisingly began the affair on top, with Palace lacking in confidence after no goals in their opening two games of the season. But Ayew, who netted once in the previous campaign, shocked the Stretford crowd when he concluded some direct play with a neat finish.
Manchester United piled on even more pressure in the second half and threatened a comeback when Scott McTominay was upended in the area. Rashford was centimeters away from scoring with his resulting penalty.
The chances that followed for United were seldom clear-cut until James notched his second goal for the club. The Welsh youngster wasn't closed down quick enough as he stepped to the right before dispatching a beautiful swerver that goalkeeper Vicente Guaita helplessly watched sail into the top corner.
Old Trafford was jubilant and could sense a miraculous rally reminiscent of Solskjaer's playing days, but it wasn't to be. The ball broke loose for Van Aanholt, and he took one touch prior to a smash that slipped under De Gea's hands.