The brilliant Blues opened the scoring through Gabriel Jesus in the first half and turned on the style in the second, inspired by the excellent Kevin De Bruyne, with further goals from Bernardo, De Bruyne, an own goal and a late Sergio Aguero penalty.
The opening 20 minutes saw few chances of any note, with Burnley well organised and defending in numbers.
Danilo also saw a powerful drive saved by Pope moments later as City started to take command and the Brazilian would soon play a hand in City breaking the deadlock on 23 minutes.
The stand-in left-back found compatriot Gabriel Jesus on the left of the Clarets’ box and the in-form striker ghosted past Kevin Long and James Tarkowski before drilling a low shot that Pope could only deflect into the roof of the net.
It was his 15th of the season and his tenth goal in cup competitions, pulling him level with Sergio Aguero as the Club’s top scorer
That was about that for a fairly uneventful opening period that the Blues dominated without ever shifting into top gear.
Burnley could - and should - have drawn level within minutes of the re-start after Nicolas Otamendi lost possession to Matej Vydra who, with just Ederson to beat, shot wide of the post.
It was a miss Sean Dyche's side would live to regret as the Blues doubled their lead just five minutes later.
And the tie was effectively put to bed in superb style just past the hour as De Bruyne collected the ball on the edge of the box before teeing himself up and hitting a sweet, unstoppable drive that arrowed in Pope's bottom left-hand corner before he even saw it.
And De Bruyne completed his man-of-the-match performance on 73 minutes as he got in behind the Burnley defence before hotting a cross that Kevin Long could do little else but fire into the roof of the net - it would be the Belgian's last act of the game as he was replaced by Phil Foden to a standing ovation.
And Long's miserable afternoon was complete five minutes from time as he fouled David Siva in the box and Sergio Aguero knocked home his 16th of the season from the resulting penalty.
What it means…
The Blues have now won eight games on the bounce and not conceded a goal in six matches - it's been 566 minutes of regulation time since Roberto Firmino scored against City on 3 January - nine hours and 26 minutes.
“I am really pleased for the commitment. Especially with all the games in our legs and minds.
“Everyone wants to win and that’s why. We do it for the people but we have done it from the beginning, since we arrived here.
“Of course, we have to improve a few things but we made good movements and I’m satisfied to be in the next round and the draw on Monday."
It’s back to Premier League action with a trip to Newcastle United on Tuesday and then the Blues take on Arsenal on Sunday 3 February at the Etihad.