Goals from Bobby de Cordova-Reid, Tim Ream and Aleksandar Mitrovic inspired the Cottagers to victory.
Marco Silva's side dominated throughout and were aided by Palace's indiscipline in either half which halted their hopes of a fourth straight home win.
Three minutes after his side went behind, Tyrick Mitchell was shown a red card for a foul on Kenny Tete, and James Tomkins followed him.
The result strengthens the visitors' European push as they are just two points off the top six.
Palace almost took the lead at the half-hour mark after struggling to impose themselves early on.
Jordan Ayew's effort crashed off the bar as he was freed inside the area.
The Cottagers paid the price moments later when De Cordova-Reid headed the visitors into a deserved lead as he pounced on slack play from Andersen before finding his team-mates with a cross.
Mitchell was thrown out by the Palace man after Tete reached a loose ball first.
There was no chance of a Palace revival after the break. Tomkins was shown a second red card for elbowing the ball off the ground.
Ream scored from a corner on the 71st minute to make it 2-0 for the Cottagers.
The referee checked the VAR monitor for a potential handball against the Serb, only to stick with his original decision, and that's when the Serbian scored.
The two red cards were the turning points of the game. Tomkins' moment of madness forced them into damage control after Mitchell's challenge on Tete.
Even though they had 11 men, they didn't do much to win the game or avoid defeat.
Their intent was never transformed into a coherent performance as they were soundly beaten.
Fears of a third successive promotion to the Championship are vanishing at a rapid rate, and they looked well at home at this level again.
They have 22 points from 16 games under Silva, six less than they got in the whole of their last season in the league.
The way in which they dominated the ball and made possession count was impressive, and they have a forward in the sort of form to make them dreams of a European place.
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