The match finished 19-all in regulation time.
The Aussies could have won the game in the first half of extra-time but failed to send the ball wide to an unmarked man with Cornal Hendricks making a try-saving tackle. They also missed a penalty early in the second-half which would have sealed the deal.
But with Sampie Mastriet receiving SA's second yellow card after going off-side, the hosts had simply too much space and scored the sudden-death match-winner with one minute, 17 seconds left on the clock.
Earlier, South Africa beat Wales 28-21 in the quarter-finals.Australia will now play New Zealand in the Cup final, while South Africa will tackle England in the third-place playoff match.
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