Stoke had Charlie Adam sent off before halftime for a foul on Wayne Rooney but defended resolutely to hold the visitors at bay until the 69th minute when Tosun scored from close range after a goalmouth scramble.
Tosun netted both goals amid a chaotic 15-minute period in the second half that was split by what seemed to be a vital Eric Maxim Choupo-Moting equaliser.
"I thought the first goal was offside".
Instead, Turkey worldwide Tosun popped up to complete his brace - and score a fourth goal in three matches - to inflict back-to-back home defeats on Paul Lambert's men, who remain mired in the bottom three.
"I think it is a hard thing for that to happen unless you are Lionel Messi or that type of player".
Caught napping on a Rooney free-kick, Adam's failure to track Tom Davies was spared by the Everton midfielder's inability to keep down an uncontested header.
Sam Allardyce is "extremely pleased" with Everton's 2-1 win against Stoke City in adverse weather conditions at the Bet365 Stadium.
Tosun, a January signing from Besiktas who has four goals in his last three games, put Everton ahead in the 69th minute and, after Stoke equalised, scored the victor in the 83rd minute with a diving header from a cross from Theo Walcott.
West Brom have now matched their club record with their seventh consecutive league defeat.
But they face an uphill battle against their Merseyside visitors, as Stoke haven't succeeded in defeating the Toffees in their last four encounters in the league.