Australia needed just 22 balls and 20 minutes of play on the fifth day at Kingsmead to secure a 118-run triumph over South Africa in the opener of their four-match series on Monday.
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.
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.
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'sPark in Port Elizabeth, with Tests to follow at Newlands in Cape Town and the Wanderers in Johannesburg.