The game opened well for both sides, and the hosts found themselves squandering a strong early chance as Chicharito Hernandez fired into the ever-approaching legs of Wayne Hennessey within four minutes. Palace's Max Meyer and Wilfried Zaha were then frustrated to break into the Hammers' box but not find anyone in yellow with their crosses.
It was a more unlikely goalscorer in James McArthur who opened the Eagles' account just minutes later however, as he raced into United's box to prod a Cheikhou Kouyate header underneath the oncoming home 'keeper, Lukasz Fabianski.
With the Eagles having taken an early lead then, Palace began to sit back as the game settled down to a more typical opening rhythm and Roy Hodgson's back four defended with resilience as West Ham launched a series of responding attacks. Pablo Zabaleta in particular kept the Eagles on their toes as he stormed frequently along their left-wing.
Lucas Perez was perhaps closest to hitting the back of Palace's net in the opening exchanges when he latched on to a dropping ball in the Eagles' box only to volley wide from 15 yards. Barely five minutes later, much to the approval of the energised home crowd, Perez then had only Mamadou Sakho to blame for his low, drilled cross not finding a man in claret and blue just yards from Hennessey's goal line when the Frenchman stood strong to send the ball ricocheting into touch.
At the other end of the pitch, Zaha found his trademark trickery nullified by a sea of West Ham legs before Aaron Wan-Bissaka roamed forward to see a probing cross well cleared by the home defence.
The Hammers swiftly resumed their bombardment of crosses, but Welshman Hennessey and his four-man defensive guard ensured the score remained 0-1. Declan Rice attempted a different path to goal, but his towering, central header from the penalty spot sailed wide in the 38th minute.
The first-half then finished in dramatic fashion, as Luka Milivojevic struck West Ham's cross-bar from a long-distance free-kick with less than a minute of time remaining.
But the drama didn't cease following the half-time break as Robert Snodgrass opened the home side's account by volleying the ball from the edge of the Palace box just inches lower than Milivojevic had managed 15 minutes earlier.
Snodgrass' strike blew the game wide open and, as the chilled east London atmosphere grew, so too did the energy of the match. Zabaleta attempted to double the Hammers' lead when he rifled an effort at goal, but again Sakho got his body in the way and sent the ball spinning out for a West Ham corner, which was dealt with by the Eagles after Patrick van Aanholt was forced to leave the field with a knock.
Attacking towards the vocal, visiting Palace fans, Zaha threaded a ball through to Meyer, but the German midfielder was crowded out in the box before, moments later, having a point-blank effort blocked by Fabianski as the ball cannoned into the area from McArthur's boot.
It was to be the hosts who were celebrating next though, after Hennessey saved well from a Snodgrass free-kick only to see Hernandez react fastest to the loose ball and strike high into the Eagles' net.
West Ham's lead was then extended in remarkable fashion when Felipe Anderson curled a ball from the right edge of Palace's box beyond the reach of Hennessey to make the score 3-1.
The Irons almost added a fourth when goalscorer Hernandez couldn't capitalise on Arthur Masuaku's cross and put the ball wide from close range, however Palace recollected themselves in time to net the game's fifth goal via Jeffrey Schlupp.
The second-half substitute bagged his second of the season with 15 minutes of the game remaining when he headed home firmly from the penalty spot to bring his side back into the game.
Zaha and Andros Townsend both then saw tame chances result in nothing before Grady Diangana sent the ball into Hennessey who deflected it over to safety.
The following 10 minutes became a tense battle, and Palace fought hard to find the game's equaliser but, frustratingly, their efforts were in vain. Referee Anthony Taylor blew his whistle for full-time to confirm that West Ham had done enough to keep the three points in east London much to the Eagles' disappointment.
Palace: Hennessey, Van Aanholt (Schlupp 55), Tomkins, Sakho, Wan-Bissaka, Milivojevic, Meyer (Puncheon 84), Kouyate, McArthur (Ayew 66), Townsend, Zaha.
Subs not used: Guaita, Ward, Sørloth, Kelly.
West Ham: Fabianski, Balbuena, Zabaleta, Anderson (Obiang 84), Snodgrass, Noble, Chicharito (Diangana 75), Diop, Masuaku, Perez (Carroll 45), Rice.
Subs: Adrian, Cresswell, Carroll, Ogbonna, Antonio.