Ilkay Gundogan added a third just before the hour-mark with Jesus missing a penalty a couple of minutes later.
The home side pulled a consolation goal back through Harry McKirdy on 79 minutes, before the impressive Cole Palmer responded with a stunning effort on 82 to complete a 4-1 victory.
Swindon Town have admirable aspirations to play their way out of League Two with attractive, easy-on-the-eye football, but it could be argued they picked the wrong team to invite a high press against by attempting to knock the ball around from the back.
In fact, both City’s first-half goals came from mistakes by the home defence who were masters of their own downfall on each occasion.
Having won the ball in the final third, Cole Palmer was fed a pass on the right and he jinked past the left-back as he ran into the Robins’ box and then played the perfect pass across the face of goal for Bernardo to tap home his eighth of the season from a couple of yards out with 14 minutes played.
The Portuguese then missed a guilt-edged chance to double the lead just four minutes later, volleying inches over with just the keeper to beat.
But the second goal came not too long after, as more Swindon over-elaboration saw Gabriel Jesus win possession outside the box before playing a one-two with Kevin De Bruyne and slotting a low shot past the keeper.
The hosts kept plugging away, creating a couple of half-chances to keep the majority of the sell-out home crowd interested, but it looked as though Joao Cancelo had killed the tie off as he hit a powerful shot that an optical illusion suggested had arrowed into the top left corner – but it had actually gone inches wide, hit the stanchion and flown across the outside of the back netting!
Within 15 minutes of the re-start, however, City had that third goal.
It was a clever set-piece from a very intelligent footballer.
Three minutes later, Bernardo was brought down on the edge of the box and referee Darren England pointed to the spot - but Jesus missed the chance to score, seeing his stuttered attempt well saved by the keeper.
That left the home side with a glimmer of hope, and when top scorer McKirdy pulled one back on 79 minutes, the home support braced themselves for a grandstand finish.
But the outstanding Palmer ended the game as a contest with a superb angled drive that capped a memorable display.
There was still time for a trademark howitzer from Rodrigo to strike the underside of the crossbar, but 4-1 felt just about right.
What it means...
City are into the hat for Sunday's FA Cup 4th round draw.
The more we see of Cole Palmer, the more reason there seems to get excited.
His confidence grows each time he plays as he glides around the pitch, and his skill on the ball and trickery caused Swindon's defence problems all night.
He tormented Robins defenders throughout the evening, assisting the opening goal with a lovely swivel of the hips and his pass selection belies his tender years.
His goal was the icing on the cake for what is a thrilling talent.
Man of the Match
Cole Palmer. Outstanding.
Subs: Carson, Slicker, Laporte, Kayky, Bobb, Egan-Riley, Mbete, McAtee, Lavia
Swindon Town: Ward, Odimayo, Iandolo, Conroy (c), Gladwin, Lyden, Simpson, McKirdy, Kesler-Hayden, Hunt, Reed
Subs: Mildenhall, Crichlow, O'Brien, Williams, East, Parsons, Dabre, Cowmeadow, Grant
With no midweek game, City are back in Premier League action a week on Saturday (January 15) as we host Chelsea at the Etihad, kick off at 12:30pm.
RODOLFO BORRELL reaction
"We were facing a big challenge in the FA Cup. It is always challenging - if you take it seriously and don't start on the front foot, you are in trouble and it can be the difference in terms of quality.
"I think we started very well on the front foot and very quickly we scored a goal, then another and could have had two or three more which is the reality.
"Towards the end of the first half we may be allowed too many counter-attacks and it was going back and forth and in the second half there were some counter-attacks again, but the game was already decided by this stage.
"But in terms of fielding a strong team, we take this competition very seriously and we are very much aware of the history of the FA Cup in this country and what it means,
"We have achieved three Premier League titles and many others, but our FA Cup win was one of our proudest."
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