South Africa dominated possession but failed to score a single try against the Wallabies.
“We know that defence wins these big games, and our discipline was really good,” Australia captain James Horwill said after the match.
It was Horwill who scored the only try of the game, in the 12th minute, to help the Australia to an 8-3 lead at half time.
Sprinboks fly-half Morne Steyn replied with a penalty and drop goal in the second half to put his team ahead, before James O’Connor kicked the winning penalty after a line-out infringement with eight minutes remaining.
In the other quarter-final, hosts and tournament favourites New Zealand overcame Argentina 33-10 in Auckland.
A determined Pumas side had gone into the half-time break just five points down, with the score at 12-7.
The Kiwis did not manage a try until 13 points before the final whistle, when Kieran Read finally grabbed a try. Brad Thorn scored a second with three minutes to go.