The match, played in humid conditions in the American Midwest, burst into life after the first hydration break, which was booed by the largely Algeria-supporting crowd.
Veteran forward Arnautovic broke the deadlock in the 28th minute with the first effort on target.
The 37-year-old controlled a long ball from David Alaba and finished calmly past Oussama Benbot, brought into the team at the expense of Luca Zidane, son of France great Zinedine Zidane.
Algeria were a whisker away from an equaliser in the 40th minute when Fares Chaibi’s powerful shot from the edge of the penalty area smashed off an upright.
But Belghali levelled in the 45th minute, dancing past three Austrian defenders before lashing home.
Algeria were fortunate that the chance came their way — defender Phillipp Mwene tried to shepherd the ball out of play but it ricocheted back off the corner flag and was picked up by Belghali.
The game fell quiet in the early stages of the second half but caught fire again in the 55th minute when Sabitzer finished off a superb team move, striking in home Konrad Laimer’s cross from the edge of the box.
There was another twist five minutes later when Mahrez finished powerfully after fine work from Houssem Aouar, who produced a pinpoint cross.
As the clock ticked down, both teams played cagily, cutting out the risks, knowing a point was enough.
But the late, late drama means Iran, who had hoped to progress as one of the best third-placed teams, are out of the World Cup.
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