The Blues started brightly and Samuel Eto'o had the first effort of the match, dragging a shot wide after turning in the box.
It was then Stoke's turn to threaten, but Peter Crouch could only direct his header narrowly wide after getting in front of his marker.
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Chelsea almost doubled their lead before the break, but Oscar smacked the post with a ferocious drive and Frank Lampard lifted his effort over the bar from 18 yards.
The hosts started the second half in positive fashion, with Schurrle hitting the post shortly after the restart. Oscar then curled an effort wide as Chelsea pushed hard for the second goal.
Stoke defended well but struggled to create at the other end, and Begovic had to be alert to keep out Lampard, before Eto'o somehow poked wide from a couple of yards.
Begovic then had to be at his very best to tip over David Luiz's free kick as the match reached its conclusion.