The Blues had gone in at the break trailing to the Clarets but dug deep in the second period with two Sergio Aguero goals in two minutes swinging the game City’s way.
Further efforts from Leroy Sane and sub Bernardo Silva sealed a hard-fought, but ultimately impressive win.
With four changes from the side that beat Watford on Tuesday, the Blues fielded a string starting line-up against Burnley.
Kevin De Bruyne was given the chance to recover from some of his battle scars with a place on the bench while Claudio Bravo and Oleksandr Zinchenko were handed rare starts.
As happened in the league meeting back in October, the Clarets defended in numbers in the opening stages and looked menacing from set-pieces.
The Blues dominated possession without making keeper Nick Pope work too hard and it was the visitors who drew first blood as John Stones’ attempt at a clearance allowed Ashley Barnes to run clear and hammer the ball past Bravo from 12 yards with 25 minutes played.
City just weren’t at the races at that point with too many passes going astray and attacks fizzling out around the Burnley box.
Zinchenko and Fernandinho sent in long range efforts but Pope was enjoying a relatively quiet afternoon as the two banks of players in front of him managed to snuff out most of the danger.
Danilo’s sliced effort just before the break pretty much summed up a disappointing first-half as it span harmlessly away for a throw-in.
Burnley had come to frustrate and some of Pope’s leisurely goal-kicks began to irk the sell-out home crowd somewhat.
But even when City aren’t quite firing on all cylinders, there is a dogged determination to find a way and just before the hour, the game was turned on its thanks to two moments of magic from Ilkay Gundogan.
With 56 minutes on the clock Danilo was fouled 30 yards from goal and as the Burnley players took their defensive positions, so Gundogan quickly took the free-kick, sliding in Aguero who made no mistake with only Pope to beat.
Two minutes later the same pair combined to put City ahead.
Gundogan received the ball on the edge of the box from Aguero who then darted past the German midfielder who back-heeled the ball into his path and Aguero did the rest, jinking round Pope to send the Etihad wild.
It was breathless stuff.
The Blues could – and should – have gone two clear just past the hour-mark as Danilo’s low shot was palmed out instinctively by Pope into Sane’s path but the young winger somehow cleared the bar from three yards out.
No matter, because the next chance he had on 70 minutes he tucked home with aplomb as the majestic David Silva slid a weighted pass into Sane’s path and he made no mistake from an acute angle.
And it was Sane who created the fourth of the afternoon, racing onto sub De Bruyne's pass before dancing round the stranded Pope and teeing up Bernardo who coolly slotted home to end the contest.
Burnley players claimed the referee hadn’t blown his whistle after Gundogan’s quick free-kick brought City level – their protestations were long and loud but referee Graham Scott was having none of it - the letter of the law says he doesn't have to play if he's been asked permission to take it.
Star man: Sergio Aguero
Instinctive finished from a natural goal-scorer who is now level with Raheem Sterling on 18 goals
What it means:
City progress to the FA Cup fourth round with the draw to be made on Monday evening and played at the end of January.
More cup action as City host Bristol City in the semi-final, first leg of the Carabao Cup.
Then it’s off to Anfield a week on Sunday as the Blues resume Premier league action against Jurgen Klopp’s Liverpool.