Posh took a deserved lead after 17 minutes when Paul Caddis needlessly gave the ball away to Grant McCann in midfield.
McCann in turn found the excellent George Boyd who turned David Prutton beautifully and slid a pass to Lee Tomlin inside the left channel of the penalty area. From 10 yards out Tomlin then hit a low shot past Phil Smith for his second league goal of the campaign.
Swindon dominated possession for most of the second half but could not turn their control into clear-cut chances until the 71st minute. Simon Ferry found Matt Ritchie in the box and the on-loan winger turned well but was brought down by Mark Little.
Austin duly stepped up and slotted home his 15th goal of the season in all competitions.
The Robins then could have won it but for two wonderful saves from Joe Lewis, denying Alan Sheehan both times from free-kicks. While with the last kick of the game, Austin connected perfectly with Sheehan's cross only to see his shot rebound off the right-hand post.