And it was the Robins who took the lead in the 37th minute as Charlie Austin grabbed his eighth goal in as many starts. The striker collected a ball from Simon Ferry and turned sharply before firing past Alan Julian.
Scott Cuthbert doubled Swindon's lead with a looping header from Jon-Paul McGovern's corner in the 52nd minute.
Substitute Vincent Pericard was then brought down by Mark Bentley and Billy Paynter fired home the resulting penalty to make it 3-0.
Chris Palmer pulled a goal back for Gillingham with three minutes remaining when he drove home a shot from the edge of the area.