Stoke was set on its way to a third straight win by Moses, who rose to power a header in from 15 meters in the 32nd minute from a cross by Phil Bardsley.
Diouf sealed victory in the 84th by converting a rebound from close range after Marko Arnautovic struck a shot against the post.
Everton, which challenged for Champions League qualification last season, has progressed to the last 16 of the Europa League this season but is in freefall in the Premier League.
Roberto Martinez's side is winless in five league matches and six points off the bottom three.
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