After Matt Renshaw's runs set the game up for Australia, Hazlewood's six wickets had South Africa all out after lunch on day five for 206 in pursuit of 421.
The seamer's 6-49 came in a typically relentless display, breaking South Africa's resistance after they survived the first 110 minutes without loss on Tuesday.
After the Proteas resumed at 4-102 and initially displayed a stoic resolve, Hazlewood was ultimately the match breaker, getting Wiaan Mulder caught down legside soon before lunch.
He then bowled Marco Jansen with the wobble seam and had Simon Harmer caught behind with a brute of a ball after the break as the Proteas fell away.
The 35-year-old's five-wicket haul was the 15th of his career, as he overtook Brett Lee into eighth spot on Australia's all-time wicket-taking list on 311.
"He gets a lot of wraps for how consistent he is, but I think that undersells him a bit," Australia's captain Pat Cummins said of Hazlewood.
"You can bowl him any time. With the new ball - class. With the older ball, he gets it reversing. Gets the speed gun up a bit. Just a captain's dream."
Aiden Markram had provided the major resistance for the Proteas, but when he was caught behind for 75 off Starc, South Africa's demise was inevitable in the battle of the world's two top-ranked nations.
Renshaw also took 2-30 with his offspinners, with his first two Test wickets backing up the 190 and 82 he scored with the bat in this Test. He was named player of the match.
Those scores were the two highest of any player from either team, setting up Australia's first-innings 390 after they had slipped to 6-124 on the opening day.
Renshaw's second-innings knock was also crucial, given it allowed Australia to push for a day-four declaration at 6-314.
The other main feature of Australia's win was their fielding.
It was evident again on Tuesday, with Alex Carey's catch diving low to his right to remove Mulder the circuit breaker for the Proteas.
Steve Smith followed it up four balls later, with a smart take at slip to a ball that came hard off Kyle Verrynne's bat from Renshaw's bowling.
That followed three superb catches from Smith in South Africa's first innings, that helped dismiss the Proteas for 284 on day three.
Smith's catch on Tuesday came after he was cleared of any concussion symptoms by Cricket Australia, having been struck on the helmet while batting on day four.
Australia's selectors will take great joy from the victory, given they were heavily criticised by the call to bring Nic Maddinson in following his 33-ball duck on day one.
But the left-hander's 48 helped set the tone in their second innings, while the decision to open with him allowed Renshaw to move down to No.5 and shine.
Australia did lead 1-0 in South Africa eight years ago after a victory in Durban, before the Proteas bounced back to win it 3-1 in one of the most dramatic series in history.
The Proteas will have strike bowler Kagiso Rabada back for the next Test in Gqeberha, but will need to improve significantly across the board.
"It's not the way you want to start," South Africa captain Temba Bavuma said.
"You have to give it to Australia. The way they dominated the majority of the game. They stuck into us even at (6-124).
"A lot of questions and conversations we need to have. We can learn from them from a batting and bowling point of view."
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