Goals from Glenn Murray and Wilfried Zaha saw Alan Pardew's side come from a goal down to earn their fifth away win of the campaign and move 11 points from the drop zone.
They now sit on 36 points - a tally good enough to guarantee survival in six of the last 11 Premier League seasons.
Mame Biram Diouf's eighth league goal of the season after 14 minutes appeared to put Stoke on course for a productive afternoon, but Zaha and Yannick Bolasie inspired Palace's comeback.
Pardew expressed a wish to overtake former club Newcastle United in the Premier League table last week, and Palace have now done that – climbing to 11th.
Palace's electric wing-play has been central to their revival under Pardew and Bolasie was quickly in the action, blazing over the crossbar, but Stoke eventually settled with Adam – one of five home changes – taking centre stage.
Having seen an earlier effort whistle just wide, Adam's 30-yard free-kick deflected off the foot of Damien Delaney inside the penalty area and Diouf capitalised to poke past Julian Speroni.
And Pardew's side turned the game on its head in first-half stoppage as Zaha latched onto a Murray flick before shrugging off the attentions of Marc Wilson and slotting home his second goal in as many games.
Full of confidence, Palace resumed strongly after the interval with Bolasie and Zaha continuing to terrorise the Stoke defence, and they went close to extending their lead after 54 minutes as Scott Dann's effort beat Begovic but came back off the post.
The visitors did not have it all their own way, though, with Speroni making a string of saves to keep their lead intact.
