The Red Devils will return to Europe's top table if Liverpool are beaten by Chelsea on Sunday, the Belgian netting the winner after Jason Puncheon equalised Juan Mata's penalty
Louis van Gaal's side took the lead midway through the first half when Scott Dann was controversially punished for a handball, Juan Mata converting the resulting penalty.
Palace grew into the game, however, and equalised after the break when Jason Puncheon's fizzing free kick took a deflection off Daley Blind on its way past David de Gea.
De Gea made a superb stop to deny Glenn Murray as the game got deeper into the second half and his resilience was rewarded when opposite number Julian Speroni's terrible late rush off his late cleared the way for Marouane Fellaini to finish off Ashley Young's cross.
Louis van Gaal once again had to make do without key United midfielder Michael Carrick, though Blind stepped into the England man's role capably and struck Speroni's left-hand post early on from 25 yards.
Palace's threat came predictably from out wide and Wilfried Zaha regularly had the beating of Antonio Valencia on his first game against United since leaving permanently in January, though the winger's delivery let him down.
The hosts gifted United the opener as Dann stooped to block Young's centre, only to be punished for handball by referee Michael Oliver – Mata stepping up to beat Speroni from 12 yards.
Headaches soon arrived for Van Gaal, however, as captain Wayne Rooney struggled through the half with a thigh issue and Luke Shaw was forced off with apparent concussion after catching a James McArthur elbow to the face, Jonny Evans replacing the teenager.
The United captain came out early for the second half in an attempt to prove his fitness but he was withdrawn before play resumed, Radamel Falcao coming on in his place.
Palace boss Alan Pardew signalled his intent by replacing Joe Ledley with Puncheon at the break and he was swiftly rewarded as the former Southampton man stepped up to a 20-yard free kick, whipping it over the wall and in via a slight nick off Blind.
Selhurst Park's fervent support ramped up a notch as Palace looked to add United's scalp to those collected against Liverpool, Tottenham and Manchester City at home this term but they found De Gea in stunning form.
The Spaniard somehow denied Murray six yards from goal before getting down smartly to keep out Zaha's volley.
And Fellaini capitalised on his defence's sturdiness, stealing in to nod home after Speroni's error to take United to the cusp of a return to Europe's top table.