It is the Owls' third win on the spin in all competitions, with a goal in each half enough to help the visitors knock out their League Two hosts.
Adam Reach opened the scoring on 27 minutes with a clinical left-foot strike after a ball over the top from Barry Bannan.
Callum Paterson sealed Wednesday's place in the next round as he slotted home from close range after a brilliant run from Liam Shaw.
The Owls youngster danced through the Grecians defence, giving Paterson the easy job of slotting the ball under the goalkeeper in the final stages of the contest.
Caretaker manager Neil Thompson was absent from the touchline as he was isolating.
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