The hosts took the lead through Mame Biram Diouf but Palace then struck twice in quick succession at the end of the first half, Glenn Murray scoring from the spot before Wilfried Zaha shot home to earn Pardew a fourth win in five away games.
Incredibly, there were no more goals in a pell-mell second-half, with Julian Speroni outstanding in the Crystal Palace goal.
And defeated Stoke were left to rue two contentious decisions from referee Andre Marriner, who gave a spot-kick against the hosts in the first-half and then denied them one of their own after the break.
Stoke made the better start and a neat back-heel from Stephen Ireland set up Charlie Adam, who put a curling effort just wide of the post.
And the hosts then took a 14th-minute lead when Adam went for a goal with a free-kick from a long way out and his low shot hit Palace defender Damien Delaney, falling perfectly for Mame Biram Diouf, who produced a cracking first-time finish, flicking the ball past Julian Speroni.
There was an element of luck in the goal but the hosts were making the running and they continued to control the game, albeit without creating any more clear-cut chances in the first half.
However, the complexion of the match changed dramatically four minutes before half-time when James McArthur floated a speculative ball forward, Asmir Begovic collided with Yannick Bolasie in the box and, to the goalkeeper’s disbelief, ref Andre Marriner pointed to the spot.
Begovic was possibly reckless in charging out but Bolasie’s foot was very high and the Stoke goalkeeper was booked for his vociferous protests.
To compound his misery, Glenn Murray brilliantly converted the penalty, lashing his spot-kick high into the net, and Palace found themselves level despite having been up against it for most of the match.
The goal seemed to unsettle Stoke, Adam and Erik Pieters following Begovic into the book in quick succession.
And Palace then scored again with virtually the last kick of the half, Murray flicking on a long Speroni punt and Zaha pouncing as Pieters dithered, before shooting past the advancing Begovic.
Inspired by their unlikely lead, Palace began an enthralling second half on the front foot, although McArthur blotted his copybook when he was booked for going down too easily in the box.
Jason Puncheon then squandered a great chance to double the lead, blazing over the bar after a Zaha cross was half-cleared.
And centre-back Scott Dann then produced a brilliant piece of skill on the edge of the box after a corner, before hitting the post with a low shot which took a slight deflection.
However, Stoke then took charge, only to find Speroni in inspired shot-stopping form.
Having already used his face to deny Diouf in unconventional fashion at the start of the half, the Palace keeper brilliantly denied Ireland after the midfielder had controlled a Marko Arnautovic shot and looked certain to level.
Another big decision then went the way of Palace when Joel Ward hurled himself in front of a Diouf effort and appeared to block with his arm. No penalty, said Marriner.
That man Speroni then came up trumps again, keeping out an Adam free-kick which had eluded everyone in the box and looked set to sneak into the corner of the net.
And, although Stoke continued to dominate possession in the final quarter of an hour and pumped plenty of crosses into the box, Palace held firm, with Dann, Delaney and company outstanding in defence.
Diouf put a late half-chance into the side netting and the visitors held out to claim yet another memorable win under Pardew, making it back-to-back league victories for just the second time this season.
Stoke: Begovic (6), Cameron (7), Shawcross (7), Wilson (7), Pieters (5), Adam (8), N’Zonzi (6), Biram Diouf (7), Ireland (7), Arnautovic (6), Crouch (6)
Subs: Walters (6)
Crystal Palace: Speroni (8), Ward (7), Dann (8), Delaney (7), Souare (6), Puncheon (6), Ledley (6), McArthur (6), Zaha (7), Murray (7), Bolasie (7)
Subs: Gayle (5), Ameobi (6), Kelly (6)
Player of the match: Julian Speroni