It was a clear equation coming into the final day, Australia needed 288 runs to win while the hosts needed 10 wickets as the series came down to the wire in what has been an epic battle between the world's top-ranked Test teams.
Kagiso Rabada gave the Proteas the almost perfect start, picking up two wickets in the very first over, having Matt Renshaw caught down by wicketkeeper Ryan Rickelton and then trapping Marnus Labuschagne.
Travis Head hit three fours to move onto 14 before Rabada snaffled his third wicket of the morning, having left-handed batter caught at gully by Marco Jansen as the Aussies were in a disaster at 22-3.
Head's scalp also saw Rabada go to the milestone of 350 Test wickets, becoming the quickest to do it in the history of the game, taking 13 808 balls - beating fellow South African Dale Steyn by 911 deliveries.
Josh Inglis was then run-out after terrible call that saw Marques Ackerman whip around from silly-point and flick the ball to Rickelton who took the stumps down with the batter short.
Simon Harmer was then introduced into the attack and the spinner had pressure of bowling his sicountry to victory on a wearing fifth day pitch.
The slow-bowler delivered the goods for his captain, getting rid of Alex Carey and Cooper Connolly in quick succession, before then castling Mitchell Starc as Australia were deep in the mire on 60-7 at the lunch interval.
Cameron Green and his captain Pat Cummins tried to shut up shop and grind after the break, only scoring two runs in 10 overs at one stage, but the resistance was never going to last.
Harmer would eventually end Green's stay at the crease, bowling number 4 through the gate as the allrounder tried to play what could only as the attacking stroke.
Cummins and Nathan Lyon took the total beyond a hundred with a partnership of 45 before the visiting skipper was lbw to Harmer, who completed his five-wicket haul.
A sign of Australia's misery was when Lyon became the top-scorer, beating Greens 27, and was 33 not out but watched on Harmer completed the victory by clean bowling last man Josh Hazlewood.
Harmer completed career-best figures of 6-16 in 21.1 overs, while it was also the first time the spinner took 10 wickets in a Test match.
The Proteas will select one of Tony de Zorzi or Zubayr Hamza to bat at number 3 in the first Test against Bangladesh, starting at the Wanderers on Sunday, according to head coach Shukri Conrad.
Australia seamer Mahli Beardman took an excellent 6/24 on his One-Day International debut as they beat England by seven wickets in Perth on Friday to take a 1-0 series lead.
Australia captain Pat Cummins has dismissed Usman Khawaja's claims that the entire team knew about the use of sandpaper on the ball in the series in South Africa in 2018 as "complete and utter nonsense".
Cricket Australia chief executive officer Todd Greenberg insists there would be "absolutely no benefit" to reopen the 2018 ball-tampering scandal investigation following Usman Khawaja's revelations.
