First-half goals from Paul Benson and Andy Williams put the League One hosts in the ascendancy on Tuesday night, and Troy Archibald-Henville sealed matters seven minutes from time after Charlie Austin had given the Clarets hope.
The visitors made the brighter start, with former Robins striker Austin going to closest to opening the scoring when he saw an early header cleared off the line by Swindon left-back Jay McEveley.
But on 19 minutes, McEveley turned provider as the home team took the lead against the run of play, his whipped left-wing cross neatly turned in on the half-volley by Benson.
Burnley continued to dominate possession - with midfielder Chris McCann and Austin both spurning decent opportunities - but it was Swindon who doubled their advantage on 41 minutes.
After a succession of misplaced passes, the ball found its way to Williams, who outpaced the Burnley defence and finished coolly past Brian Jensen.
Austin eventually marked his return to the County Ground with a scrappy tap-in on 75 minutes, but Archibald-Henville's 83rd-minute header - his first goal for the League One side - secured the win.