Rory Donnelly and Josh Wright were denied by Robins keeper Lawrence Vigouroux, while Bradley Dack hit the side netting for the hosts.
Jonathan Obika had an effort blocked at the other end before Donnelly then headed straight at Vigouroux.
John Egan also went close for the Gills when he fired wide but Swindon held on to avoid a third 1-0 defeat in a row.
Swindon failed to score for a third consecutive game as they drew at promotion-chasing Gillingham.