Tom Soares put Palace ahead against the run of play, heading in Scott Sinclair's right wing cross with 21 minutes gone.
Stoke missed a number of chances before Jose Fonte doubled the Eagles' lead on the stroke of half-time.
Glenn Whelan pulled a goal back for the home side with five minutes gone, but it was a case of too little too late.
After the game, Stoke boss Tony Pulis said: "We didn't get the luck we needed tonight. The header from Mamady Sidibe which hit the inside of the post would have gone in a few weeks ago.
"We had lot of chances to get back in the game and it just wouldn't go in, but I saw enough to stay confident."