The Clarets were second-best throughout the encounter as Claude Puel's impressive side scored three times in 14 second-half minutes.
Saints were on top from the kick-off and Tom Heaton somehow managed to keep Austin's fifth-minute header from Matt Targett's cross out before Dusan Tadic fired a long-range effort straight at the Clarets goalkeeper.
Saints, who had kept six consecutive clean sheets in all competitions before the game, controlled the early stages of the encounter, although Sam Vokes' 20th-minute effort sailed wide of Fraser Forster's right-hand post as the visitors finally began to get a foothold in the match.
Jeff Hendrick then lashed an ambitious effort over the bar midway through the first half while, on the half-hour mark, Heaton was forced to make a smart stop to deny Tadic.
Heaton saved Nathan Redmond's rasping effort, while Burnley had penalty appeals waved away by referee Mike Dean when Johann Berg Gudmundsson went down under Virgil van Dijk's challenge as the first half ended goalless.
Southampton deservedly took the lead seven minutes after the restart when Austin toe-poked the ball over the line from close range after van Dijk had caused chaos in the Burnley defence by winning a header.
Saints doubled their lead on the hour mark when a corner found its way through to Redmond who lashed a low drive past Heaton and, five minutes later, it was 3-0 when Austin sent the Burnley goalkeeper the wrong way from the penalty spot after Gudmundsson had brought Sam McQueen down.
Sean Dyche's side did manage to end Southampton's long run without letting in a goal when Vokes stroked a 72nd-minute spot-kick home after Tadic had conceded a soft penalty when tussling with Ben Mee, but the visitors were well beaten on the day and are now only one point above the relegation zone.