The 2-1 victory over the Cherries was only the second in their last 22 league games and while the South Coast side still sit in the relegation zone, they are now just a point from safety and have 17th-placed Swansea City firmly in their sights.
Southampton drew 1-1 with Everton on Saturday evening to keep the Premier League relegation battle very much open.
The remainder of the half though was largely a non event with only Austin being yellow carded just before the break for pulling down Seamus Coleman.
With the numerical advantage and few structural and formation changes, the Merseyside club took used it to good effect and equalised at the death after Tom Davies's shot deflected into the net to deny Southampton their survival with two games still to play.
In the 96th minute, Southampton's Ryan Bertrand kept the ball in bounds, but Everton's Idrissa Gueye collected it near midfield and sent it up to Davis at the top of the area.
See the joy and despair for either side by clicking on the video of the goal above.
Charlie Austin eventually raised eyebrows around a muted Goodison Park, the striker's instinctive first-time effort from 20 yards drawing a smart diving save out of Jordan Pickford.
To say the game was low on incident and excitement would be a considerable understatement.
Sam Allardyce continues to divide opinion, the vast majority of which seems intent on seeing him removed from the Merseyside club. The school of science, it is being said by some unhappy supporters, is presently in special measures. The passion they showed at the end of the match last weekend also shows exactly the type of mentality they will need to carry into their current games before they end their season in Manchester against a City side brimming with momentum. He does not exactly court popularity, though something needed to change and after an anonymous first half the winger could hardly complain.
Moss awarded a foul towards Nathan Redmond, who had given Southampton the lead, after he appeared to drag again Everton defender Ramiro Funes Mori.
Everton, unbeaten in their previous four games, had a chance in the 66th minute but the timing was off on Oumar Niasse's cross to Cenk Tosun at the far post. Cue joy in the small corner of the Bullens Road Stand that was red and white and more boos from the blue majority.
However, they looked to have weathered a late storm, with Alex McCarthy making a fine save to thwart Leighton Baines.