City, under the guidance of assistant coach Rodolfo Borrell after Pep Guardiola contracted COVID-19, never looked in danger of suffering a first FA Cup defeat to a side from the fourth tier or lower since 1984.
The visitors were cruising by the half-hour mark, with Bernardo Silva and Gabriel Jesus getting the goals.
Ilkay Gundogan buried a fine free-kick with just under an hour played to put the game beyond doubt, before Harry McKirdy got a consolation.
Cole Palmer's gorgeous late strike finished off the League Two side and capped a great performance late on.
Swindon was lucky to hold out for 14 minutes, but Silva, who had previously wasted two great chances, was never going to miss when the excellent Palmer presented him with a tap-in.
City then pounced on Louis Reed's questionable flick just outside his own area as Swindon attempted to play out from the back, with Kevin De Bruyne feeding Jesus into the box for a simple finish.
The contest was over, if it wasn't already, in the 59th minute as Gundogan cleverly curled his free-kick around the wall and into the bottom-right corner.
The Swindon goalkeeper's wall positioning may have played a part, though Lewis Ward went some way to making amends when saving Jesus's penalty after Jordan Lyden fouled the Brazilian.
City could not keep a clean sheet, with Jonny Williams capitalising on Rodri's error and playing McKirdy into the area to finish past the suspect Zack Steffen.
But the away side had the last laugh, Palmer planting an exceptional finish into the top-left corner.