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The Black Cats were dominant in the early stages, Jack Rodwell blasting an effort wide after just a few minutes played.
The deadlock was eventually broken by the home side in the 16th minute when Jozy Altidore turned on the edge of the box and smashed the ball into the bottom corner past Potters goalkeeper Jack Butland.
On the half-hour mark, however, Stoke equalised thanks to Muniesa, whose effort was knocked over the line by Costel Pantilimon's leg.
After the break, it took both sides a while to get into the game, with neither goalkeeper being tested, but in the 62nd minute, Butland had to make a great save to keep the scores level, diving down low to his left to palm away Will Buckley's side-footed effort.
The away side went on to take the lead midway through the second half, as Muniesa picked up a loose ball in the area following a botched Sunderland clearance, before powering an unstoppable shot past Pantilimon.
The deflated Black Cats were unable to come back into the game, and could have conceded another late on when Marko Arnautovic was left unmarked, but the Austrian volleyed over from close range.