Charias Mavrias's goal after five minutes was enough for the Black Cats to book their place in the next round.
The Greek winger gave the home side the lead when he drilled a low shot into the bottom corner after he managed to escape the attention of a couple of challenges.
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Mavrias was involved again on 25 minutes when he set up Emmanuel Giaccherini, but the Italian could only flash his shot over the bar inside the area.
El-Hadji Ba had an effort for the Black Cats after the restart but it was comfortably held by Lewis after coming through a crowded penalty area.
Santiago Vergini nearly marked his Sunderland debut with a goal, but he headed wide at the back post with 15 minutes remaining.