A late Marouane Fellaini winner saw United return to winning ways and all but seal a return to the Champions League with a massively important away victory.
Juan Mata had earlier put the Reds in front but we were pegged back by Jason Puncheon's deflected free-kick before Fellaini headed in Ashley Young's cross 12 minutes from time to secure a hard-earned win.
United moved seven points clear of fifth-placed Liverpool, who travel to champions Chelsea on Sunday knowing that anything but a win would mathematically end their top four hopes.
In a first half of relatively few chances, Julian Speroni saved well from Daley Blind and Ander Herrera lashed an effort wide before United were given a penalty when Scott Dann handled a Young cross.
Mata put United's recent spot-kick woes to bed by converting from 12 yards for his ninth of the season but it was a bittersweet half for the Reds when we lost both Luke Shaw and Wayne Rooney to injury.
Palace were much improved after the break, with Puncheon stepping off the bench to bend home a free-kick via a deflection off Blind in the United wall shortly before the hour mark.
That led to a period of sustained pressure from the hosts.
James McArthur poked wide and United were then indebted to David De Gea who made a brilliant point-blank save to deny Glenn Murray as wingers Yannick Bolasie and Puncheon started to dominate the game.
However, it was the impressive Young who had the final say when his superb delivery from deep picked out Fellaini who nodded in after Radamel Falcao had put the home defence under pressure.
Palace were not finished there, though, and called De Gea into action again when he produced another superb stop to keep out Murray in the closing stages but United held out to move to the brink of Europe's elite competition.
United Faithful Man of the Match: A close call between Young and De Gea but we're giving it to DDG for his two superb stops from Murray that proved so pivotal.
Overall team performance: 7.5/10