Gabriel Jesus opened the scoring in the 23rd minute, before Bernardo Silva, Kevin De Bruyne, an own goal from Kevin Long and a Sergio Aguero penalty made it five for the Premier League champions in the second period.
City's first serious attempt of the match arrived in the 16th minute when Kyle Walker picked out Bernardo Silva inside the Burnley box, but Pope did brilliantly to keep out the latter's low strike.
The Burnley stopper then kept out a powerful attempt from Danilo as the home side started to find their groove.
The pressure was relentless, however, and City made the breakthrough in the 23rd minute when Gabriel Jesus worked some space inside the Burnley box before smashing a powerful attempt through the body of Pope.
Burnley were having their moments on the counter, particularly through the impressive Dwight McNeil, but the Premier League champions continued to create openings late in the first period.
VAR had a look at a potential penalty in the 33rd minute when Gundogan went down under a challenge from Burnley defender Charlie Taylor, but play continued without the award of a spot kick.
Gundogan had another attempt deflected wide of the post late in the first period, before John Stones headed the resulting corner past Pope's goal as City's one-goal lead stood at the half-time interval.
Burnley held their own for long spells of the opening 45 minutes, but it was a different story in the second half as City flexed their muscles once again.
City, who have not won the FA Cup since 2011, had more than 75% of the ball in the first period, but it was Burnley that started the second half on the front foot and they should have levelled the scores in the 48th minute.
It should have been 1-1 early in the second half, but it was instead 2-0 to City in the 53rd minute, with Bernardo Silva's low strike from inside the Burnley box being deflected past Pope in what was a decisive moment.
Pope had to keep out a powerful Gabriel Jesus strike just before the hour, before Riyad Mahrez had one deflected wide moments later as the pressure from the home side continued.
City's third arrived in the 61st minute and it was a sweet strike from De Bruyne, who found the bottom corner with a wonderful effort from outside the box.
Burnley defender Long then turned a De Bruyne cross into his own net in the 73rd minute to make it four for the Premier League champions.
The hosts then had their fifth in the 85th minute when Aguero converted form the penalty spot after Long had pulled David Silva's shirt inside the Burnley box.
City, who have now won their last eight matches in all competitions, will return to Premier League action with a trip to Newcastle United on Tuesday, while Burnley will take on in-form Manchester United at Old Trafford on the same night.
MANCHESTER CITY (4-3-3): Ederson; Walker, Stones, Otamendi, Danilo; Fernandinho (D Silva 65'), Gundogan; Mahrez, De Bruyne (Foden 75'), Bernardo; Jesus (Aguero 75')
BURNLEY (3-5-2): Pope; Long, Tarkowski, Gibson; Brady (Cork 46'), Hendrick, Defour (Westwood 77'), Taylor, Ward; Vydra (Wood 62'), McNeil