Said Benrahma had fired the Irons' into the lead against the run of play with a 20-yard screamer but the West Ham defence undid all his good work with a defensive balls-up five minutes before half-time, allowing Wilfried Zaha to haul Palace level.
Then, in the fifth minute of stoppage time, Michail Antonio squandered a good opportunity with a weak cross straight into the arms of Palace keeper Vicente Guaita.
Just 17 seconds later the ball was in the opposite net after a Palace breakaway with Michael Olise's strike deflecting in off Aaron Cresswell.
Bookies reacted to the dreadful manner of the defeat by slashing Moyes odds to be the next manager to be sacked from 20/1 to 10/1.
Only Jesse Marsch (7/2), Frank Lampard (9/2), Steve Cooper (6/1) and Thomas Frank (7/1) sit above Moyes in that market.
The tide has been turning against Moyes with the West Ham fanbase for some months now, with only their European performances papering over the cracks.
Since beating Liverpool 3-2 last November, West Ham have won just 13 times in 41 league games, mostly against sides struggling at the wrong end of the table.
His team selections have been somewhat baffing too, sticking with the out-of-form Tomas Soucek and shuffling one of West Ham's few impressive players Said Benrahma in and out of the action.
Moyes' decision to substitute Benrahma against Palace was met with a chorus of boos, while removing of the misfiring Soucek was loudly cheered.
"I thought he (Benrahma) scored a good goal, but I don't know if he had much of an impact after," said Moyes after the game. "There's a lot of experts out there, isn't there?"
Now the West Ham board have come out to back their under fire boss.
"We one hundred per cent support the manager," co-owner Davis Sullivan told Hammers News regarding question marks over Moyes' future.
That kind of comment has never been a good thing for a manager skating on thin ice.