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It didn't take long for the breakthrough to come, however, as a Charlie Adam free kick on the quarter-hour mark crept through the Palace wall, where Mame Biram Diouf broke away to seize the loose ball and fire home.
Stoke continued to threaten as Julian Speroni spilt Adam's lofted cross from the right wing moments after the restart, but Diouf was unable to grab a second goal from the mistake.
The visitors were awarded a penalty five minutes before the break when Asmir Begovic brought Bolasie down in the box, with Murray smashing high into the roof of the net from the spot to get back on level terms.
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Just as it seemed like it would be all level at the break, the Eagles turned the game on its head when Wilfried Zaha used his pace to power through and finish with precision.
Stoke had a good chance to level proceedings after the restart through Diouf, but the ex-Manchester United man's shot was stopped by the face of Speroni in unorthodox fashion.
The woodwork denied Palace a third minutes later as Scott Dann's low shot found its way through a crowd and against the bottom of the right upright.
Diouf was in the thick of it once again midway through the second half as his effort towards goal appeared to hit the hand of a Palace player in the box, but nothing was given by the man in charge.
The action showed no signs of letting up as Speroni then pulled off another smart save, this time preventing Steven Nzonzi's flick going into the net from Adam's delivery.
Palace were able to survive the tense latter stages of the game to secure a valuable three points away from home and climb to 11th in the Premier League table, while Stoke slip to ninth.