The Robins just about shaded the first half and Billy Paynter gave them the lead after 31 minutes, converting a penalty after he had been blatantly pushed by Jim McNulty.
They were only ahead for seven minutes as Elliott Bennett's superb right-wing cross was headed powerfully in by Nicky Forster.
Brighton had started brightly without creating clear chances but Swindon kept Peter Brezovan busy. The goalkeeper did well to collect a thunderous shot from Jonathan Douglas and Austin saw a shot dropped just eight yards out.
Swindon continued to attack but they might have conceded when Tommy Elphick won a free-kick but McNulty's effort was deflected wide.
Douglas might have restored Swindon's lead but Brezovan did well to punch away, before Austin headed home Jon-Paul McGovern's superb cross for a 54th-minute winner.