Clarets boss Sean Dyche hailed a "sublime" finish from Steven Defour as Burnley eased through to the fifth round of the FA Cup for the first time since 2011.
Defour laid on the opener for Sam Vokes right on half-time - his 50th goal for the club - and added a glorious second after the break, lobbing keeper Fabian Giefer to knock Championship side Bristol City out.
And Dyche said: "In my time here the fans have been fortunate to see some good stuff and successful football, but that was a sublime finish.
"If it was in the Premier League they'd be showing it a thousand times.
And he was delighted with the performance, which earned a sixth-successive home win: "We controlled a lot of the first half.
"We scored a very good goal just before half-time and obviously that was a very important goal as well because we had been on the back foot a little bit.
"And then in the second half we were in good form and I thought we came out worthy winners in the end but that's no disrespect to Bristol because they made it tough and they played well and that's why I was pleased."
Dyche made eight changes to the side, with James Tarkowski starting in the centre of midfield, and Dyche noted: "I said in the last round I enjoyed the way the team played and I think today there was a lot of positive things.
"There's players who just let you know that they are ready all the time. It's a great mentality.
"I mentioned in the last cup game how important the players who are not always in the side are and here they are vital because they look after themselves and they're ready to go when called upon."
Burnley are looking to bolster their squad ahead of the transfer window closing on Tuesday, but missed out on Hull City wideman Robert Snodgrass, who chose to join West Ham, after both the Clarets and Hammers had £10m offers accepted: "Everyone has to agree. I spoke to him only briefly and I think a decision was made quite quick. West Ham are a big club with strong financial power."