Arnautovic scored in the seventh and 15th minutes from assists by Xherdan Shaqiri as Stoke exploited poor defending by City, which was without Sergio Aguero, Yaya Toure and Vincent Kompany because of injury.
It was City's third loss in its last five games in all competitions — and fourth of its league campaign — to fuel concerns about the team's shortcomings when its star players are missing.
Playing expansive football at times, Stoke handled the windy conditions better and could easily have won by a bigger margin. Arnautovic hit the post when put clean through for his hat trick.
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