The match marked Toffees manager Ronald Koeman's first return to the club he left for Merseyside in the summer, and he watched his ex-side totally outplay his current one for the majority of the 90 minutes.
However, efficiency was once again the issue for the Saints, meaning the game was far closer on the scoresheet than the balance of play.
Nevertheless, the result pushes Southampton back into the top half of the Premier League, while Everton are still searching for their first away win since early September.
Spectacular start from the Saints
Manager Koeman's last words to his team before taking the pitch would surely have been something along the lines of keeping it tight early and getting a foothold in the game as the away side.
However, the Everton players did just the opposite. After the Toffees gave possession away almost straight from the kick-off, a slick passage of play from the home side eventually led to a Cedric Soares cross from the right. The visitors could not deal with the wicked delivery, and the ball found its way into the path of Southampton's 19-year-old debutant Josh Sims, whose skewed effort went across goal and presented top scorer Austin with the simplest of headers, putting the hosts ahead within the first minute.
Fifteen minutes later, the youngster Sims, who did not look out of place at all, arguably should have doubled the Saints' lead, but he could only meet Nathan Redmond's precise cross with a tame header from eight yards out which hardly troubled Everton goalkeeper Maarten Stekelenburg.
Claude Puel's side's dominance was showing no signs of letting up when, midway through the first half, an impressive run and pass from midfielder Pierre-Emile Hojbjerg found Austin in the penalty area. However, the Englishman's low effort presented Stekelenburg with another routine save.
Game levels out
Everton finally showed some attacking impetus moments after Austin's chance, as Seamus Coleman's low cross found Idrissa Gueye, but the midfielder blazed his attempt over the bar from just behind the penalty spot.
Koeman's side continued to grow into the game and they worked another decent opportunity just after the half hour, with Gareth Barry's looping header dropping onto the roof of Saints goalkeeper Fraser Forster's net.
In spite of this resurgence, Southampton went into the break with a deserved 1-0 lead.
Home side resume dominance
The second half began in the same way that the first did, with Hojbjerg striking the post from a tight angle after the ball found its way to him from a corner. Leaving his waiting team-mates, who clearly believed after a pass back across goal was the better option, disgusted.
The Toffees grew into the game quicker this time around however, but Romelu Lukaku and Yannick Bolasie each missed good opportunities with wayward efforts.
This entertaining encounter kept on giving. Stekelenburg was forced into a fine save to deny Austin a second headed goal, before Lukaku appeared to be brought down unfairly by Virgil Van Dijk in the penalty area. Despite this, referee Craig Pawson was unmoved.
With 15 minutes to go Southampton's James Ward-Prowse was presented with a glorious opportunity to perhaps seal the three points for the hosts, after Austin recovered a fumble from Stekelenburg and squared it to the midfielder. However, he could not make any meaningful contact. The-22 year-old came close again soon after, when he was denied by a combination of Stekelenburg and Phil Jagielka from close range.
As the final whistle began to loom, Everton pushed forward with greater intent, but besides half-chances for Enner Valencia and Gareth Barry, the Saints defence was relatively untroubled.
It would have been a travesty if the men from the South Coast had not won today on the balance of play, but win they did, closing the game between themselves and Everton to just two points and returning to the top half.