The Hammers, who had made their worst start to a season since 1988-89, scored the only goal of the game 19 minutes in through Manuel Lanzini's well-taken strike and were able to hold on despite being reduced to 10 men late on.
Palace struggled to really create many clear-cut openings but did have the chance to level up from the spot on the brink of half time, which Christian Benteke failed to make the most of as he missed the target altogether.
It means a first defeat in six for the Eagles, who were aiming to level a club-record unbeaten run this evening, but a vital victory for their opponents as they climb out of the drop zone and into 15th place.
The visitors experimented with a different five-at-the-back system in an attempt to wake up from their slump, and it appeared to work early on as they bossed possession and picked up the ball in some dangerous positions.
Despite their slow start to the game, Palace were still finding gaps at the back as a result of the returning Aaron Cresswell and makeshift wing-back Michail Antonio springing up the field, but Wilfried Zaha dragged his shot wide when picked out in acres of space.
United made the breakthrough with a little under a quarter of the match played after some patient build-up play, which culminated in Cresswell sending in a cross for matchwinner Lanzini to delicately place past Mandanda.
That was the Argentinian ace's third goal in the same number of games against the Eagles, and a goal that West Ham deserved on the basis of their play up until that stage in the game, but they struggled to truly push on for the remainder of the half.
A Lanzini shot aside, which Mandanda did well to get behind in the wet conditions, the following 20 minutes proved to be bland with neither side able to truly muster any openings.
The match finally burst back into life shortly before the interval, though, as Angelo Ogbonna tripped up Benteke inside the box, seconds after Antonio had done likewise on Zaza, to give the Belgian a chance to level up from the spot.
A terrible penalty followed; Benteke sent his attempt wide of the target, before skimming the post with a headed attempt moments later to ensure that his side went into the break without testing Adrian on a single occasion.
Alan Pardew turned to Zeki Fryers and Yohan Cabaye at the interval in an attempt to turn things around, looking to prolong his side's unbeaten run, yet things continued in the same manner as Palace struggled to find any moments of quality.
United were instead looking to make the most of their rare opportunities when they arrived, one of which fell their way 56 minutes in when Mandanda raced out to close the angle on Lanzini, who instead picked out Dimitri Payet.
Payet's own attempt was easily dealt with, and that may have acted as the motivation the Eagles needed to finally find some momentum in the game - Andros Townsend sending in his side's first on-target shot of the match two minutes later and Zaha being denied a free header at the back post by some good defending.
Another quiet spell in the game soon followed, however, which suited West Ham more so than their opponents, who were simply lacking inspiration to make something happen in a congested final third of the pitch.
Jason Puncheon tried his best to spur his side into life, sending in a rather tame attempt for Adrian to collect, but a big moment in the contest was to arrive with 15 minutes to play.
Antonio first saw his header cleared off the line by Puncheon, before Cresswell went down too easily under minimal contact and earned a yellow card from Martin Atkinson.
Sixty seconds later the full-back was again in trouble for halting Zaha's run, with Atkinson showing him a second yellow card to make it four red cards in four outings between these two sides.
Benteke and Tomkins both came close to finding a leveller in the remainder, while Adrian needed to produce a fine stop to deny Connor Wickham at the death, as the Hammers made it three wins here on the bounce.