The Australian victory advances the club to the final match on Sunday against the hosts, India.
Australia managed to bowl out South Africa for a below-par 212 despite David Miller's defiant 101 at the Eden Gardens. The chase, however, was not as straightforward as Pat Cummins and his team would have liked.
Man-of-the-match Travis Head (62) led the team's rollicking start before a mid-innings wobble put Australia in a tough spot.
South Africa did not let big partnerships bloom, and their inadequate total meant Australia eventually prevailed with 16 balls to spare to make the final match for the eighth time.
"Nerve-wracking couple of hours but great effort and a great game," said Cummins, who was especially pleased with his side's fielding. "It's something we talk a lot about. It wasn't up to scratch at the start of the tournament."
The chase has not been South Africa's forte in the tournament, so Temba Bavuma's decision to bat first under an overcast sky seemed to be a no-brainer.
Bavuma played the match despite not having fully recovered from a hamstring strain and his four-ball duck suggested sitting out would probably have been a wiser decision for him.
Australia ramped up the pressure with their electric fielding and Cummins led by example, taking a brilliant catch to send back the other South African opener, Quinton de Kock.
The winners were so dominant that South Africa had to wait until the ninth over for their first boundary and were 18-2 after their 10 power-play overs.
It soon snowballed into a full-fledged crisis and South Africa slumped to 24-4 in the 12th over.
Drizzling rain held up play after 14 overs, and when the action resumed Heinrich Klaasen (47) and Miller rebuilt the innings, taking South Africa past the 100 mark in 28 overs.
Klaasen hit Adam Zampa for back-to-back sixes, but Australia regained control after part-time spinner Head broke the 95-run stand, claiming two wickets in two balls.
Miller brought up his hundred with a six off Cummins but departed in the same over trying to clear the rope again.
In contrast, Australia got off to a flyer, plundering 60 runs from the first 10 overs with David Warner smashing four sixes in his 29.
Bavuma introduced spin in the seventh over and the move immediately paid off as Aiden Markram bowled Warner with his first delivery.
Kagiso Rabada dismissed Mitchell Marsh in the next over but South Africa's catching was rather sloppy and Head was the beneficiary on two occasions either side of his fifty.
Just when Australia appeared to be cruising to a comprehensive victory, South Africa's spinners injected fresh excitement into the contest. Keshav Maharaj bowled Head through the gate and Tabraiz Shamsi dismissed Marnus Labuschagne and Glenn Maxwell to jolt Australia out of any complacency that might have crept in.
Steve Smith made 30 and Josh Inglis scored 28 but South Africa kept fighting back.
However, Cummins and Mitchell Starc combined to foil the Proteas and get Australia over the line.
"Quite hard to put into words," a dejected Bavuma said. "Our character came through. It was a dog fight.
"The way we started with the bat and the ball was probably the turning point, we lost it quite badly there."
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