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.
Palace, who had struggled to trouble their opponents, then took the lead through Marouane Chamakh who 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.
Charlie Adam then warranted his inclusion in the starting XI with the equaliser after good work from Crouch.
Begovic maintained Stoke's advantage with a great save from Jason Puncheon after Adam had given the ball away in the middle.
Gayle fizzed a volley wide but Stoke held out and secured the three points.