The Addicks took the lead in controversial circumstances midway through the first half, with Swindon players furiously protesting that Paul Benson had handled in the build-up to Johnnie Jackson's deflected opener.
The visitors were back on level terms after 41 minutes as Rob Elliot failed to hold on to Matt Ritchie's header but the midfielder reacted quickly to stab the spilled ball past the Addicks keeper.
A mistake by Charlton captain Christian Dailly gifted Swindon a second goal 11 minutes after the restart, with Austin taking full advantage to fire the visitors in front from 18 yards.
Sean Morrison headed home Jon-Paul McGovern's 77th-minute free-kick before Austin added Swindon's fourth goal four minutes later with a header after losing his marker.
Pawel Abbott pulled one back for the hosts with a neat turn and shot in the 87th minute but it was scant consolation for the home fans, who vented their anger at manager Phil Parkinson at the final whistle.