Austin shot low into the net from 12 yards after a ball into the box from substitute Mark Marshall was only half cleared by the Brighton defence.
Brighton started well and winger Kazenga Lualua saw a deflected shot go just wide inside the first minute.
Swindon lost keeper David Lucas early on with a leg injury and Brighton midfielder Andrew Crofts put a low shot just wide of stand-in keeper Phil Smith's left hand post midway through the first half.
Brighton had a strong penalty claim waved away when Glenn Murray, who earlier had a header cleared off the line, went down under a challenge from skipper Gordon Greer.
Source: Team Talk
Source: Team Talk