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Just before the half-hour mark, City made it two through Gabriel Jesus, who calmly slotted in after being played through on goal by Kevin de Bruyne.
With the clock ticking close to 60 minutes, Gundogan, skipper on the night, added a third with a wonderful curling free-kick into the bottom corner.
Minutes later, Jesus missed a golden chance to make it four, seeing his spot-kick saved. The hosts then pulled a goal back with just over ten minutes remaining as Harry McKirdy raced through and fired past Zack Steffen.
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Swindon had an extra spring in their step after the goal, but Palmer quietened the home crowd with a sensational finish which cannoned off both bar and post.
View the best images from the clash as City booked a place in the fourth round.