The visitors required a single wicket to secure success after South Africa resumed on 293 for nine, chasing an improbable victory target of 417.
Five runs were added by the home side before Quinton de Kock was hit on the back pad and trapped leg before wicket by Josh Hazlewood (3-61) as he tried to launch the ball over the boundary.
Mitchell Starc earlier had a chance to complete a hat-trick with the first ball of the day after being denied by bad light on Sunday but bowled down the leg side to De Kock, who was eventually out for 83 after adding two to his overnight tally.
Morne Morkel was not out on three as Starc (4-75) finished with nine wickets in the match.
"It was a nice way to start, said victorious skipper Steve Smith afterwards. "The first test in a series is always important and the way we played was a real plus."
Australia's win has been overshadowed by a heated off-field exchange between players at the tea interval on the fourth day, which the International Cricket Council (ICC) are investigating, as well as a charge for spinner Nathan Lyon.
CCTV footage from the players tunnel showed Australia vice-captain David Warner and home wicketkeeper De Kock involved in a fiery exchange as players climbed the stairwell to their dressing rooms during the break.
The video shows Warner being restrained by Australia batsman Usman Khawaja and spin bowler Lyon as he directs a verbal barrage at De Kock, who left the field as South Africa's not-out batsman at the same time as the Australians.
"It's rather unfortunate that the incident took place and certainly not in the spirit of the game," South Africa's team manager Mohammed Moosajee said in a statement.
"The match referee had spoken to us after the day's play. We will await any further communication from the ICC around this incident."
In a separate incident, Lyon dropped a ball on prone batsman AB de Villiers after he was run out prior to lunch on Sunday as South Africa's top order collapsed in their run chase.
Lyon faced an ICC sanction after being charged by match referee Jeff Crowe, officials said on Monday.
It sets up an intriguing rematch in the second test, which starts on Friday at St George's Park in Port Elizabeth.
"There is no doubt South Africa will come hard at us in Port Elizabeth but hopefully our boys will be up for the fight," Smith added.
(Reporting by Mark Gleeson; Editing by John O'Brien/Peter Rutherford)
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