The Hammers were handed a free kick just after the half hour as Abdoulaye Faye fouled David Di Mechele.
Spanish striker Diego Tristan took responsibility and his right-footed effort made keeper Sorensen wish that he was just a few centimeters taller. Instead, the ball sailed past the keeper's stretched arms for 0-1.
Stoke did try to make amends for that goal but West Ham held on to their lead for a narrow but vital victory.
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