Jay Rodriguez hit four goals as Burnley needed extra-time to see off the challenge of League Two side Burton Albion.
The England Under-21 striker put the Clarets 2-0 up with a brace of penalties before Cleveland Taylor responded for Burton.
Charlie Austin made it 3-1 with seven minutes left but Calvin Zola and Jaques Maghoma forced extra-time.
Ross Wallace put Burnley back ahead and Rodriguez added two more.
After a goalless first 45 minutes, Rodriguez opened the scoring from the spot after Austin was fouled by Tony James.
He doubled the lead with another spot-kick after Wallace had been impeded by Adam Bolder, but Burton halved the deficit through Taylor.
Austin's spectacular long-range effort looked to have won the game but Zola and then Maghoma struck in the dying stages.
Wallace took less than a minute of extra-time to hit a thumping 25-yard drive and then two minutes later Rodriguez made it 5-3, firing in a superb strike from 30 yards out to compete his first senior treble.
The 22-year-old then added his fourth of the night to complete the scoring following a one-two with Wade Elliott.
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Burnley boss Eddie Howe told BBC Radio Lancashire:
"These are hard games, people will look at them on paper and think well you should cruise it but football is never like that.
"We are thankful we are not on the end of a shock result and we progress to the next round which is the aim.
"We'd like a home tie, that would be ideal but you cant read to much into it we'll just see who we get."
Burton boss Paul Peschisolido told BBC Radio Derby:
"We kept going in the 90 minutes, when we countered we looked more dangerous but our final pass wasn't there.
"Credit to them they're a good side but I'm proud of the boys."