The visitors were reduced in number after 55 minutes when Alex Pritchard suffered a rush of blood and pushed over Wade Elliott after being fouled by the City midfielder, who received a yellow card for his part in the incident.
From then on the home side dominated. But Wes Foderingham saved well from Jay Emmanuel-Thomas and Sam Baldock, while centre-backs Troy Archibald-Henville and Jack Stephens excelled.
City goalkeeper Frank Fielding had to make a fine double save from Michael Smith after 57 minutes and it ended up a frustrating afternoon for his improving team.
An even first half saw Jay McEveley shoot narrowly wide for Swindon and Baldock head over the crossbar for City from a Greg Cunningham cross.
The save of the match came from Foderingham who dived to keep out Elliott's low shot after a slick City move midway through the first half.
Massimo Luonga went close for the visitors with a low shot across the face of goal which passed just wide of the far post.