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However, minutes later Howard was powerless to prevent Moses from putting the Potters ahead as the forward guided Phil Bardsley's cross into the top corner.
In the second half, the Toffees pressed to get an equaliser as Romelu Lukaku took the ball in stride on the edge of the box and his powerful strike looked to be heading on target until Geoff Cameron deflected the ball wide.
Roberto Martinez's men came close when Leon Osman drilled a strike towards goal, which was blocked into the path of Arouna Kone, who poked a shot towards the net, but Begovic was on hand to make the stop.
Everton were dominating possession as they attempted to force their way back into the game, but were struggling to break down the resolute backline of the Potters.
The home side's resilience paid off as they secured their win in the 84th minute when Marko Arnautovic broke through the Everton defence and fired an effort off the post, only for Mame Biram Diouf to knock in the rebound.