Two goals in six first-half minutes from Fernando Torres and Ramires gave the visitors the win, although their success was tempered by the loss of Marco van Ginkel and the Brazil international midfielder to injury.
Torres opened the scoring in the 29th minute with a poacher’s finish after Juan Mata saw his shot saved by Wes Foderingham.
And Ramires doubled the lead soon after as he lifted a ball into the net following a neat pass from Torres.
Jose Mourinho gave Mark Schwarzer his Chelsea debut in goal, while there were recalls for Mata and David Luiz. Cesar Azpilicueta made his 50th appearance for Chelsea, while there were also starts for Kevin De Bruyne and Willian.
Swindon boss Mark Cooper made one change from the side victorious against Bristol City last time out, with Alex Pritchard replacing Louis Thompson in midfield.
Chelsea were forced into an early change when young Dutchman Van Ginkel twisted his knee after falling awkwardly, with the 20-year-old appearing to be in a lot of pain as he hobbled off in the 10th minute.
After a tight opening to the game, Mourinho’s men had the first sight of goal midway through the half.
Mata threaded a neat ball into Torres and the striker cut in front of his man but saw Foderingham make a save with his head to deny him.
Minutes later though the former Liverpool man made no mistake as he broke the deadlock after neat build-up play between Ramires and Mata.
Mata’s shot was saved by Foderingham, but Torres was on hand to pounce on the loose ball and poke home.
And Torres then turned provider in the 35th minute as the visitors doubled the lead.
He played a perfectly weighted pass for Ramires to run onto inside the penalty area, and the 26-year-old calmly lifted the ball over the advancing keeper him and into the back of the net.
Danny N’Guessan thought he had halved the deficit for the hosts early in the second half when he met a Jay McEveley free kick and headed past Schwarzer, but the goal was ruled out for offside.
Swindon were providing more of an attacking threat after the break, but Schwarzer was rarely tested in the Chelsea goal.
And the hosts were nearly punished for pressing forward in the closing minutes but Torres and Willian were both denied by resilient defending as the Premier League side eased through to round four.
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