Stoke were reduced to 10 men on the half-hour as Charlie Adam was sent off for a reckless challenge on Wayne Rooney.
Tosun scored his first of the afternoon on 69 minutes, reacting quickest after Michael Keane's effort had been blocked.
Stoke substitute Eric Maxim Choupo-Moting equalised on 77 minutes, poking the ball home from a free-kick.
But, seven minutes later, Tosun headed in Theo Walcott's cross to claim the points.
Everton stay ninth with 40 points, while Stoke remain 19th on 27 points, four from safety.
Stoke: 1 Apr v Arsenal (A)
Everton: 31 Mar v Man City (H)