Jonathan Walters scored twice for the Potters in extra time as it finished 1-1 between the two teams after 90 minutes.
Stoke took control of proceedings at the Britannia Stadium thereafter, but failed to create any clear-cut chances.
Palace had a great chance to take the lead in the 34th minute when the referee awarded them a penalty for Ryan Shawcross's foul on Wilfred Zaha. However, Jermaine Easter saw his spot kick saved by Thomas Sorensen.
The Championship side started the second half without Zaha, who was taken off at half time with an injury, and had defender Danny Gabbidon to thank for his timely blocks on the hour mark.
Palace levelled the score with three minutes of normal time remaining, as Shawcross brought down Williams inside the area and Glenn Murray successfully converted the penalty.
However, Stoke had the last laugh as Walters headed home Etherington's cross at the back post in the fifth minute of extra time before firing home his second on 110 minutes.
Cameron Jerome added his name to the scoresheet when he found the target in the dying moments of the game from close range.