Given their respective positions in the FIFA World Rankings, the hosts were strong favourites to register a comfortable victory.
They struggled to exert any kind of dominance, though, with the only real chances in the opening exchanges falling to Philipp Lienhart and Michael Gregoritsch, who both missed the target.
It took until the 24th minute for either goalkeeper to be tested, but Patrick Wimmer’s tame effort from an unlikely angle was easily saved at the near post.
Stefan Posch should then have done better with a close-range header that went wide, before the talismanic Gregoritsch, comfortably the top scorer in Ralf Rangnick’s squad with 24 international goals, was forced off through injury.
His replacement did not take long to win a penalty, though, as Sasa Kalajdzic was about to pull the trigger when he was scythed down in the area by Idan Nachmias.
Romano Schmid was on spot-kick duties, and he duly sent goalkeeper Daniel Peretz the wrong way with a low effort that put his team 1-0 up.
Posch could then have doubled the lead in first-half injury time, only for Peretz to make a vital save.
Lienhart missed with another close-range header after the break, and that wayward finishing proved costly as Sayed Abu Farchi levelled for Israel with a thunderbolt header that went over the dive of Alexander Schalger and nestled in the back of the net.
Joy turned to anguish within moments though, as the Israel forward - booked just minutes earlier for pushing at a corner - was shown a second yellow card after picking up the corner flag during his celebration.
That gave the hosts a huge lift, but for a solid save by Peretz down to his right, Kalajdzic would have made it 2-1 with a low strike from outside the area. The towering striker then forced another stop out of the goalkeeper as he met a cross from the left with his head, before Paul Wanner was also thwarted.
A draw looked a certainty until Mwene met Wanner’s pass at the back post and confidently converted.
Moments later, with Israel chasing the game, Kalajdzic got the goal his performance deserved to put the icing on the cake and extend Austria’s unbeaten home record against Israel - relegated back to League B for this campaign - to six games.
Flashscore Man of the Match: Sasa Kalajdzic (Austria)