A magnificent goal from Steven Defour capped another Turf Moor win as the Clarets cruised through to the fifth round of the FA Cup for the first time since 2011.
Sam Vokes scored his fifth goal in the competition and 50th in total for Burnley to set Sean Dyche’s in-form side on their way to a place in Monday night’s last-16 draw.
And Defour’s second-half masterpiece then made sure of avoiding any chance of an upset to the Championship strugglers as the Clarets secured a sixth successive home win.
Defour arrived at Burnley in the summer with a reputation as a class act after winning almost 50 Belgium caps and league titles in two different countries.
And in one breath-taking moment he showed why with a stunning finish – lifted over goalkeeper Fabian Giefer as Burnley broke from one end of the pitch to the other - that will long live in the memory of those in an impressive crowd of 14,921 who were there to see it.
The return to fitness of Scott Arfield and Johann Berg Gudmundsson – both absent since the Clarets’ goalless draw at Sunderland in their first round-three meeting – increased Dyche’s options as Burnley looked to extend their remarkable home run.
But with Dean Marney out following the season-ending injury he sustained at Arsenal and Jeff Hendrick given a breather, there was a central midfield vacancy for James Tarkowski to fill.
And the big defender looked perfectly at ease as he helped the Clarets largely control the first half without being too threatening in front of goal before Vokes struck with the last kick of the first half.
Tarkowski directed one long-range effort over the bar and Arfield also saw a shot blocked at the end of an excellent move triggered by Ben Mee and continued by Defour.
But the Championship side was never overly stretched as they kept the Clarets at arm’s length and tried to counter-attack themselves in search of a morale-boosting win and place in round five for the first time since 2001.
Lee Johnson’s side reached the last 32 with a replay win at Fleetwood Town but had lost their last eight league games to slip within two points of the drop-zone.
With that in mind, Johnson started with top scorer Tammy Abraham on the bench and towering striker Milan Djuric leading the Robins’ attack.
And the former Cesena frontman, signed earlier this month, should have put the visitors ahead with their best chance of the first half.
Zak Vyner’s right-wing cross was delivered perfectly for Djuric to meet with a header from eight yards out but the Bosnia and Herzegovina international was unable to beat Nick Pope, who spread himself to save well.
And three minutes later Vokes struck to set the Clarets on their way.
The striker can do little wrong in the FA Cup and he added to his collection when he swept the ball first-time inside the far post after Defour had won possession, played a one-two with Gudmundsson and then wriggled his way into the penalty area before picking Vokes out with the perfect pass.
With their noses in front the Clarets were almost helped to put the tie to bed seven minutes into the second half when Aiden Flint’s mis-placed back pass almost caught out Giefer, whose toe-end saved major embarrassment as he just managed to get the ball out for a corner.
Defour then went close as he cut inside to curl an effort just wide but that finish would have paled into insignificance alongside the quality of the goal he did score to make it 2-0.
City had just sent on Abraham as they sensed they might nick a goal to set up the possibility of a replay when Burnley showed their Premier class to put the game beyond them with 22 minutes to ogo.
Pope beat out a free-kick at one end to give possession to Defour, via a little pass from Gudmundsson, who advanced over half way before accepting another return pass from the Icelander.
The midfielder still had plenty to do as he approached the penalty area, but with the most delicate of right-footed chips lifted the ball over the lanky Giefer with a finish matching precision and beauty to claim his second Burnley goal.
Vokes then hit the outside of a post as he missed the chance to add his sixth goal of the season.
But the Clarets had plenty in hand as they stayed in perfectly in the groove and firing on two fronts ahead of Tuesday night’s visit of Premier League champions Leicester City.
Burnley: Pope, Darikwa, Keane, Mee, Flanagan, Tarkowski, Barton (Hendrick 70), Arfield (Barnes 70), Defour (Kightly 78), Gudmundsson, Vokes. Substitutes: Robinson, Lowton, Ward, Gray.
Bristol City: Giefer, Bryan (Abraham 61), Flint, Brownhill, O'Dowda, Golbourne, Reid, Djuric (Engvall 61), Vyner, Hegeler (Pack 76), Wright. Substitutes: Fielding, Wilbraham, Paterson, Magnusson.
Referee: Andre Marriner.