Former QPR and Burnley striker Austin, who netted a first-half double against Sparta Prague in the Europa League in midweek, came off the bench to score the game’s only goal shortly after the hour mark.
His first Premier League goal since his debut winner against Manchester United in January, Austin controlled the ball with his chest before lashing home from eight yards.
Southampton move up to 14th on five points, while Francesco Guidolin’s side, who performed poorly on the south coast, remain without a win since the opening day of the season.
They were without defender Neil Taylor, conspicuous by his absence following his public fall-out with manager Guidolin, who subbed him before half-time against Chelsea last weekend.
But it was in attack where they laboured, with summer signing Fernando Llorente never looking like opening his goal account.
Southampton bossed proceedings from the off and were almost gifted the lead when keeper Lukasz Fabianski spilled a shot from Shane Long.
Steven Davis and Dusan Tadic charged in to tuck away the follow-up only to get in each other’s way and put the ball behind, but fortunately for the duo an offside flag spared their blushes.
Long thought he should have been awarded a penalty after going down under Ferderico Fernandez’s challenge in the area, but referee Mike Jones saw it differently and booked the Saints forward for a dive.
Long’s mood did not improve when Nathan Redmond opted to shoot rather than square the ball across goal and blazed his effort high and wide.
And Redmond was guilty of an even worse miss moments later when he collected Tadic’s low cross and stepped inside Jordi Amat, only to scoop his shot over from eight yards out.
The one-way traffic continued after the break with Long nudging the ball past the onrushing Fabianksi, only for Kyle Naughton to slide in and clear before Tadic could tap into an empty net.
Long then did put the ball into the net but Redmond had been flagged offside as he pulled the ball back.
Long’s afternoon was over shortly after but Austin’s first action was to crash Redmond’s perfectly-weighted ball across the area onto the crossbar.
However, Austin made no mistake in the 64th minute when Tadic’s cross was inadvertently helped on by Amat, chesting the ball at the far post before hammering his volley past Fabianski.
Swansea could still have snatched an unwarranted point when the ball dropped to Gylfi Sigurdsson in the area, but Fraser Forster palmed away his shot to ensure Saints got their league season up and running.