Chris Greenacre was the hero. His fourth goal of the season in the 58th minute decided the contest, but it also angered Swindon boss Andy King, who felt he was offside.
Greenacre volleyed home from a flick on by Danny Harrison following a cross by Sam Aiston.
Rovers were unlucky not to double their advantage four minutes later when Harrison rattled the crossbar with an excellent strike from the edge of the box.
It was a poor first half, with Swindon packing midfield and employing just one striker in a containing system.
But they had to thank keeper Tom Heaton for excellent saves from Iain Hume and Greenacre.
Heaton also did well after half time to deny Hume on two occasions, but he had luck on his side in the 56th minute when he headed clear at the edge of the box into the path of Steve Jennings, whose shot was blocked by Patrick Collins.
Collins was outstanding in the back four for Swindon, while Ian Goodison was outstanding for the winners.
Swindon changed their system for the closing stages, but rarely looked like gaining the breakthrough.