A late Zlatan Ibrahimovic strike earned United a third successive victory and kept up the pressure on the teams above them.
On a night of tension, controversy and drama - and a string of shocking refereeing decisions - James McArthur's strike cancelled out Paul Pogba's controversial opener.
That looked to have cruelly denied the Reds a deserved victory until our two showpiece summer signings, Pogba and Zlatan, linked up as the former found the latter to lash home a superb strike two minutes from time.
Palace were left raging at referee Craig Pawson's decision to reprieve Marcos Rojo after the defender's late, two-footed lunge on Wilfried Zaha.
United's opener also looked to have a hint of controversy about it, too, as Zlatan looked to have handled the ball into the path of a suspiciously offside looking Pogba who knocked in from close range.
The Reds were also left angered by the official, too.
Juan Mata's perfectly good goal was chalked out for offside, Joe Ledley's handball in the penalty area went unpunished and Yohan Cabaye did not receive a second yellow for a late challenge on Michael Carrick.
United started brightly against a strangely passive Palace side and went close twice early on.
Wayne Rooney headed narrowly wide and Pogba went close from distance.
Wayne Hennessy saved well from the outstanding Pogba and then also kept out Juan Mata before the visitors moved ahead in first half stoppage time.
Maya's free-kick was flicked on by Zlatan - with a hint of handball and Pogba, who looked offside, bundled in from six yards.
Despite that setback, Palace started the second half on the front foot and immediately upped the ante.
Half-time substitute Joe Ledley flashed an effort wide and David De Gea had to be at his best to deny McArthur from distance.
Palace were knocking on the door and pushed in open on 66 minutes when Damien Delaney's sublime flick played in McArthur who, this time, got the better of De Gea.
Undeterred, United responded well and almost restored their advantage instantly.
Chasing Sir Bobby's record, Rooney forced a superb save from Hennessey before another moment of controversy.
Mata thought he had scored when he tapped in Rojo's header across goal only to be denied by the linesman, and then Ander Herrera's header looked to have been blocked by Ledley's hand inside the area.
Substitute Jesse Lingard was denied by the increasingly impressive Hennessey, but then came the late sting in the tail.
Zlatan latched on to Pogba's superb clipped pass, took the ball on his chest and lashed home on the angle with two minutes left to spark wild celebrations in the away end.
Overall team performance: 8/10
United Faithful Man of the Match: Paul Pogba.