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Stoke had the majority of the early play and forced a succession of corners. However, they were unable to force Speroni into action.
Crouch then failed to hit the target with a header from six yards out from a cross from the left from Etherington.
Palace, who had struggled to trouble their opponents, then took the lead through Marouane Chamakh as he nipped in to tap home from a long ball.
Stoke almost equalised straight away as Crouch headed against the bar, before the ball hit Speroni who just managed to save.
Palace debutant Jose Campana went close to making it 2-0 before the break but his free kick went just over.