After four league games without a victory, the Saints were seeking to avoid their worst start to a season since 2000-01 and got the break on 63 minutes when Austin chested down a deflected cross and fired an unstoppable shot into the top-right corner.
Southampton dominated the game and Shane Long, Nathan Redmond and Cedric all missed good chances to give them an earlier lead.
Austin, who had scored twice against Sparta Prague in the Europa League on Thursday, could have grabbed a second after good work by the influential Dusan Tadic but his shot was well-saved by Lukasz Fabianski.
The win moved Southampton up to 14th place, one ahead of Swansea.
(Reporting by Neil Robinson, editing by Alan Baldwin)
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