Everton manager Koeman left Southampton during the close season after two seasons at St Mary's, during which he guided the south coast club to their best Premier League finishes of seventh and sixth.
But the manner of the Dutchman's departure meant he was given a frosty reception by the home supporters.
Austin's early header -? the striker's ninth goal of the season -- added to Koeman's growing concerns about the form of his side, who have now won just one of their last eight Premier League games.
Southampton could have registered a more convincing win, but victory at least meant they could put the disappointment of Thursday's Europa League defeat by Sparta Prague behind them.
They climbed to 10th, while Everton remain seventh.
Koeman could hardly have imagined his reintroduction to the south coast would have gone quite so badly, with his side falling behind inside the opening minute.
The visitors had barely had a kick when Cedric's cross flew over visiting skipper Phil Jagielka before bouncing into the path of Josh Sims via the chest of Everton right-back Seamus Coleman.
Sims, a 19-year-old winger, was making his Southampton debut, but showed impressive composure to lift the ball back across the face of goal for Austin to head home from close range.
Koeman has yet to win over Everton's supporters after his new side's impressive start to the season began to falter.
Everton's performance during the opening half was never likely to convince those doubting the new manager's conviction that a significant turnaround is about to happen any time soon.
It did not help Koeman that having criticised Jagielka recently for conceding three penalties, the veteran England centre-back played an unwanted part in Southampton's goal.
Everton booed off
But Jagielka was not alone in appearing shaky as Southampton threatened to wrap the game up inside the opening 25 minutes.
Had Sims shown the same composure he had done in the first minute, he might have directed a free header past Maarten Stekelenburg instead of steering his effort straight at the home goalkeeper.
Nathan Redmond and Pierre-Emile Hojbjerg were both guilty of sending shots over the bar when they should have found the target.
Koeman is attempting to introduce a higher tempo game to a group of players who had lost direction under previous manager Roberto Martinez.
But they struggled to impose themselves on Southampton before Idrissa Gueye shot wastefully over from 12 yards after being picked out by Coleman's clever pull-back.
They would have reached half-time on level terms if Gareth Barry had kept a far-post header down from Leighton Baines's free-kick.
The boos that greeted the half-time whistle from the visiting supporters spoke volumes about the difficulties Koeman and his players are currently enduring.
They returned for the restart several minutes before their hosts, but there were few signs of a fresh urgency to their play as Southampton quickly assumed control of the second period.
Hojbjerg grazed the woodwork in the 49th minute when he should have pulled the ball back for Austin, while the striker's header was later pushed away by the diving Stekelenberg.
In between, Romelu Lukaku miscued a direct free-kick, underlining Everton's lack of threat in front of goal.
It took the continued excellence of Stekelenburg to prevent Southampton adding to their tally.