Ashley Barnes added to his list of high-profile strikes but four second-half goals killed the Clarets’ hopes of an FA Cup shock as all-conquering Manchester City extended their incredible unbeaten run.
The Premier League leaders are now undefeated in domestic football this season, with their only loss in 38 games a dead Champions League rubber against Shakhtar Donetsk.
City were made to work for their latest success after Barnes had fired the Clarets in front in the 25th-minute to add to the equaliser he bagged here three seasons ago and the goals he has scored at Old Trafford, Anfield and White Hart Lane.
But Burnley haven’t won in 16 attempts at City – stretching back to 1963 – and that record was extended as Pep Guardiola’s expensively-assembled side went up a gear after the break to cruise into the fourth round once they had levelled with a controversial equaliser following a quickly-taken free-kick.
Sergio Aguero was the beneficary as he led the fightback with a rapid-fire double before Leroy Sane and substitute Bernardo Silva added to the tally to match the three-goal winning margin City also enjoyed in the league meeting between the sides in October.
Both sides made four changes from their last Premier League line-ups, with the Clarets bringing in Matthew Lowton, Kevin Long, Ashley Westwood and Sam Vokes.
City, despite the absence of Kevin De Bruyne, had no shortage of big hitters in their side, despite Guardiola’s protestations at their Christmas holiday workload.
And inevitably the league leaders started on the front foot with Sane and Aguero threatening to pick holes in the Clarets’ back line.
But Burnley stood firm and as they began to press and force errors at the other end, the tide began to turn,
Barnes - invariably coming in dangerously from a left-sided role - was only inches away from opening the scoring after 18 minutes when he failed to get a touch on a towering header back across goal from ex-City youngster Ben Mee following a Johann Berg Gudmundsson corner.
But the in-form striker made no mistake with a thunderous finish to put the Clarets 1-0 up just seven minutes later with his fourth goal of the season.
John Stones’ mis-kick from a Vokes’ flick-on left Barnes in the clear and after racing into the penalty area he lashed a right-footed shot past Claudio Bravo’s left hand to spark great scenes of celebration among the 7,600 Burnley fans packed into the top two tiers of the south stand.
Despite their wealth of possession City were denied any clear sighters of goal as the Clarets defended with great discipline.
And with the ever-present Jack Cork and Westwood, starting his third cup tie of the campaign, working tigerishly in front of them there were openings for the Clarets on the break.
Five minutes into the second half Barnes sent an effort over the top but in the blink of an eye the tie was ripped from the Clarets’ grasp with two goals in as many minutes from Aguero.
City’s record scorer was the scourge of the Clarets last season with three of the four goals that secured a Premier League double and the Argentina international did it again.
His first will have infuriated the Clarets as they were caught out by a quick free-kick referee Graham Scott allowed Ilkay Gundogan to take.
Aguero raced unchecked into the penalty area to slot the ball past Nick Pope and the same combination then opened up the visitors’ defence again with a one-two completed by Gundogan’s back-heel allowing Aguero to go around Pope and roll in his 18th goal of the season.
It could have been three moments later when Sane missed from close range after Pope had saved from Danilo.
But any hopes the Clarets had of making him pay with a late fightback were dashed when the Germany made international made amends by making it 3-1 with 20 minutes to go.
This time David Silva was the architect with a perfectly weighted through-ball for Sane to take a touch and then steer a low shot into the far corner.
City had the luxury of bringing on De Bruyne for the final quarter of an hour and he immediately made his mark as City then added a fourth with eight minutes to go.
De Bruyne’s pass released Sane and with Pope chasing a lost cause, the winger squared the ball for substitute Bernardo Silva to fire home with his first touch after coming off the bench to keep City on course for a possible league and cup four-timer.
Burnley: Pope, Lowton, Long, Mee, Taylor, Gudmundsson (Walters 75), Cork (Wells 68), Westwood, Barnes, Hendrick, Vokes. Substitutes: Lindegaard, Marney, Defour, Ulvestad, Bardsley.
Booked: Westwood, Cork
Manchester City: Bravo, Danilo, Stones, Otamendi, Zinchenko (Walker 72), Fernandinho, Gundogan (De Bruyne 75), Sane, D Silva, Sterling, Aguero (B Silva 79). Substitutes: Moraes, Walker, Mangala, Toure, Diaz.
Referee: Graham Scott