Led by a powerful forwards display, the Wallabies squeezed out the five-point win on the back of a second-half try to outstanding centre Pat McCabe and three penalty goals to fullback James O'Connor.
The morale-booting win over the most experienced Springboks team in Test history sets the stage for a potentially tournament-deciding showdown with the All Blacks in Brisbane in two weeks.
Desperate to atone for last week's 30-14 drubbing in Auckland, Australia had standouts across the park, with besieged skipper Rocky Elsom silencing his doubters with a storming performance.
Australia's triumph soured veteran Springboks captain John Smit's 105th and last Test in his home city.
And it was the big away scalp the Wallabies craved before heading to New Zealand for the World Cup next month.
South Africa led 6-0 at halftime but, despite only six points being scored, it was a frenetic, action-packed opening 40 minutes.
Francois Steyn drilled a 48-metre penalty goal in the second minute after Wallabies captain Rocky Elsom was booked on a touch judge's report before Butch James made it 6-0 in the 20th minute after the Australian scrum was pinged.
Aided by a lopsided penalty count, the Springboks enjoyed a territorial advantage but the Wallabies looked far more likely on attack on the rare occasions they had the ball.
Fullback Kurtley Beale went with completing an exciting long-range raid from the Wallabies after the halftime siren had sounded, but coach Robbie Deans would have been relieved to have his side in the sheds still in the contest at the interval.
The Boks could easily have been some way in front, with James missing an easy penalty goal attempt, then skewing a shot at drop goal, while the video referee took an age to deny a try to Jaque Fourie in the 25th minute.
Quade Cooper erred in letting a box kick from Boks halfback Fourie du Preez bounce close to the Australian in-goal but fortunately for the Wallabies the TV match official deemed the South African centre had knocked on.
It didn't take long for the Wallabies to hit the front after the break, with O'Connor slotting a penalty in the 44th minute before putting McCabe over for his first Test try shortly after.
O'Connor missed the conversion and the Springboks regained the lead through another James penalty in the 58th minute.
Some brilliant defence from replacement centre Anthony Faingaa allowed Australia to nudge ahead 11-9 thanks to a 38-metre penalty goal from O'Connor.
Faingaa, not long on the field for Adam Ashley-Cooper, rushed up on JP Pietersen and forced the penalty when the Boks winger was unable to release the ball.
Despite driving rain lashing into his face, O'Connor calmly struck the difficult shot between the uprights.
Another O'Connor penalty six minutes from time gave Australia a little more breathing space and the Wallabies defended their lives out as the Boks attacked relentlessly in the tense final stages.
