After being set up by Matt Renshaw with the bat, Josh Hazlewood's career-best haul of 6-49 helped Australia have the Proteas all out 206 after lunch on day five in pursuit of 421.
Australia have recorded one bigger overseas win this decade, when they beat a woeful Sri Lanka by an innings and 242 runs last February without Cummins.
But Tuesday's first Test success in Durban was a triumph against a Proteas side ranked No.2 in the world, who had not yet lost any of Temba Bavuma's 12 Tests as captain.
And not least of which after slumping to 6-124 on day one, before Renshaw's 190 helped bat Australia to dominance with the help of Cummins and Nathan Lyon.
"It's just a great way to kick off the series," Cummins said.
"It felt well behind where we wanted to be on day one. And then mainly through Renshaw, clawing our way back and grinding it out.
"Then once we had the lead, not giving it back.
"Even sometimes where we have got that far in front, I feel like we haven't been as ruthless as we could. But our standards were very high."
Australia's win kicks off an epic trilogy of overseas tours to South Africa, India and England in the next 10 months, on which this team's legacy will no doubt be judged.
But this was not your typical Australian victory of recent times.
They showed both gears, with Australia's run-rate of 2.83 in the first-innings 390 the slowest Australia has scored in a total that big in 14 years.
And after bowling South Africa out for 284, Renshaw's 82 helped Australia's 6(dec)-314 come at their fastest ever run-rate of 5.41 for a total of beyond 300.
"I think you just need to sometimes just grind it out, wait for the game to come to you and we just missed those moments in the past," Cummins added.
"That's us as a group, as we look at the next evolution."
Australia's selectors will take great joy from the win, given the heavy criticism they received following Nic Maddinson's 33-ball first-innings duck.
But his shock recall as opener netted a 48 in the second innings, and allowed Renshaw to move to No.5 and shine.
Ultimately, this Test will be remembered for Renshaw.
He made the top two scores of the game and followed it up with his maiden two Test wickets with the ball.
Steve Smith's three great catches on day three will also endure, with he and Alex Carey securing two more vital ones to break the back of South Africa before lunch on day five.
The work of Australia's bowling also shouldn't go overlooked.
They adjusted well to pull back their lengths on day three when the Proteas got away to a flyer, before Hazlewood shone as the man on day five.
His 6-49 came in a typically relentless display, breaking South Africa's resistance after they survived the first 110 minutes after resuming at 4-102 on Tuesday.
Hazlewood, who had struck twice on the previous afternoon, first had Wiaan Mulder superbly caught by a diving Carey down legside on 29.
Then after lunch he bowled Marco Jansen (35) with the wobble seam, and followed it up with a brute to catch Simon Harmer's glove on the way through to Carey.
Hazlewood'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.
And when Mitchell Starc had Aiden Markram caught behind for 75, South Africa's demise was inevitable.
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," Bavuma said.
"You have to give it to Australia. The way they dominated the majority of the game ... A lot of questions and conversations we need to have."
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