Cork climbed highest to head home a Brady cross 27 minutes in, before the Irishman rifled into the roof of the net from an angle to get his own name on the scoresheet.
Blackburn registered just one attempt on target all evening as they struggled to force their way back into the game, giving the Clarets a safe passage through to the next round.
Relive how the 90 minutes of action unfolded with Sports Mole's live text coverage below.
STARTING XI: Raya, Williams, Smallwood, Feeney, Chapman, Whittingham, Mulgrew, Ward, Caddis, Antonsson, Gladwin
SUBS: Leutwiler, Samuel, Graham, Evans, Bennett, Conway, Doyle
STARTING XI: Pope, Taylor, Cork, Tarkowski, Barnes, Brady, Westwood, Walters, Bardsley, Long, Arfield
SUBS: Heaton, Lowton, Mee, Vokes, Wood, Defour, Gudmundsson
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Tony Mowbray: "Ultimately the passion and the emotion of the game is from the supporters. They live here, work with people who support Burnley, and have to go to work the next day and either suffer or poke fun at someone on the other half of the divide. It is not just another game, the players go to Sainsbury's to do their shopping and you might get stopped by people who recognise you and they tell you how much it means to them."
Sean Dyche: "I like [Mowbray], I like him a lot. You try not to forget the people who have done right by you and when I started out at Watford, and I had a really tough start, we played them when he was manager of Middlesbrough at the time. We played really well and got beaten 1-0, and he was fantastic afterwards in his press comments. I've never forgotten it because I was getting a lot of stick at the time and I thought we'd played well on the night. You don't forget those little things. And I think he's a proper bloke as well."
BLACKBURN ROVERS SUBS: Leutwiler, Samuel, Graham, Evans, Bennett, Conway, Doyle
BURNLEY SUBS: Heaton, Lowton, Mee, Vokes, Defour, Gudmundsson
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