Swansea's afternoon got off to the worst possible start as Jonathan Walters battled hard to win the ball and lifted it into his own goal.
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The home side began to have more of the ball, but Stoke were doing a good job of closing them down.
Stoke's excellent start then got even better as they doubled their lead with 25 minutes gone.
Just before that goal, Wilfried Bony had a decent chance for Swansea, but volleyed wide.
Begovic was called into action before the break, denying Bony as the striker attempted to go around him.