Koeman's men had won just one of their last seven league games heading into the encounter and the Dutchman was desperate to get back to winning ways against his former club.
Koeman a miserable start to his afternoon as he was given a hostile welcome by the Saints fans before Austin dealt his old boss a major blow with the opener within the opening minute.
Everton created a number of chances to level, but Idrissa Gueye in particular wasted a fine opportunity as Southampton held on for a narrow, yet deserved, win
The Goodison Park side remains seventh in the table after this weekend's result with 19 points from 13 games, but its hopes of finishing in the top four to qualify for Champions League football continue to fade, with fourth-placed Arsenal nine points clear.
The host, meanwhile, is up to 10th place in the table, just two points off Everton.
Southampton got off to a flying start and Austin handed his side the lead after just 41 seconds. Debutant Josh Sims did well to set up the striker after having his initial shot blocked and Austin had no trouble whatsoever to find the net from close range.
The home side got chances to double the lead via Sims and Austin, but the former's header after a good cross from the left was straight at Everton goalkeeper Maarten Stekelenburg, while the goalscorer's low attempt after a quick attack lacked the power to worry the Dutchman, who spent last season on loan at the Saints.
Gueye should perhaps have levelled in the 26th minute after a superb team move involving Gareth Barry and Seamus Coleman, the holding midfielder wildly blasting a shot over the crossbar after the latter's cutback.
Barry then got a chance to equalise when he headed just over after a free-kick from the left, while Ross Barkley also failed to get his header on target in the closing stages of the first half after Yannick Bolasie's cross.
Southampton went in search of a second goal after the interval and Pierre-Emile Hojbjerg was unfortunate not to add his name to the scoresheet when he hit the upright from a narrow angle after dancing past Bolasie.
Austin was next to threaten for the home side with a sublime header after a cross from the right, forcing Stekelenburg into a superb save to keep his team's chances of getting a result alive.
Bolasie looked dangerous for Everton on two occasions, but the former Crystal Palace man's aim was off twice as Southampton held on for a narrow 1-0 win, with James Ward-Prowse missing two late chances to net a second.