Burnley 2, Bristol City 0
Goals from Sam Vokes, right on half-time, and a glorious lob midway through the second half from Steven Defour saw the Clarets through,
Sean Dyche made eight changes to the side, with the fit-again Scott Arfield and Johann Berg Gudmundsson back, while Nick Pope, Jon Flanagan, Tendayi Darikwa, James Tarkowski, Joey Barton and Vokes also came in.
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Hide AdCity started brightly, belying their Championship form, which has seen them lose a club-record eight games on the spin, with Milan Djuric lifting an effort high and wide after getting in behind Michael Keane.
Arfield pulled the ball across the face of goal after a lovely ball from Darikwa, and Burnley started to dominate possession, without producing a clear opening.
Tarkowski fired over from 25 yards with time and space, while Gudmundsson couldn't quite get a shot away from a Defour pull back.
Three minutes from the break, Pope was forced to pull off a fine stop to beat away a free header from Djuric from a Zak Vyner cross, and right on half-time Burnley made the Robins pay.
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Hide AdDefour won possession and linked with Gudmundsson, taking the return pass and riding a challenge before pulling the ball back for Vokes, who guided his finish inside the far post.
In the second half, Defour fired just beyond the far post, before, after Pope punched a Josh Brownhill free kick clear, Gudmundsson fed the Belgian, who supplied a sumptuous finish over Fabian Giefer and in.
Vokes hit the post from a Defour free kick as Burnley searched for a third, but the Clarets eased home and await Monday night's draw for the last 16