Charlie Adam equalised in the 58th minute before Ryan Shawcross grabbed the winner at Britannia Stadium to give Mark Hughes his first win as Stoke manager after taking charge over the summer.
Shawcross had been partly at fault for Palace's 31st-minute opener as the England centre-back got muscled off the ball by Morocco striker Marouane Chamakh, who cut inside and scored his first goal since joining from Arsenal in the off-season and his first in the Premier League since September 2011.
Stoke opened the season with a 1-0 loss at Liverpool, while Palace lost 1-0 to Tottenham.