Given initially saved Leighton Baines's penalty after Ashley Williams fell under pressure from Phil Bardsley, only for the ball to bounce back off the post and deflect over the line off the goalkeeper's head.
The home side enjoyed the best of a thrilling match and both Romelu Lukaku and Williams could have given Everton the lead during the first half, the latter denied a debut goal by Peter Crouch's goalline clearance.
Stoke responded well to going behind, Marko Arnautovic going closest when his effort rattled the bar in the 58th minute.
(Reporting by Ed Dove, editing by Ed Osmond)
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