Set 232 to win, the Proteas could only manage 224-9 in their 50 overs, leaving the series tied at 1-1 with the decider in Hobart on Sunday.
Finch did neither one thing nor the other, and chopped a delivery from Pretorius on to his stumps after scoring 41.
But a lionhearted performance by Stoinis and the Australian pacemen ensured victory despite some belligerent hitting from South Africa's lower order.
The South Africans slipped to 68-4 inside 15 overs before David Miller (51 from 71 balls) and captain Faf du Plessis rallied.
Quinton de Kock, Aiden Markram, Heinrich Klaasen and Dwaine Pretorius were all guilty of gifting their wickets to the Australians, three of them with poor shots, and in Markram's case, another bad piece of running which saw him short of his ground after he'd made 19. The teams play the second of three matches on Friday in Adelaide, with the third ODI set for the Gold Coast in Queensland state on Sunday.
After 35 overs, the scorelines were still neck and neck, with South Africa having reached 5 for 162, and when Pretorius spliced a pull tamely to midwicket shortly before the match entered the final 10 overs, the set-up was virtually identical to what had transpired in Australia's innings.
South Africa won the toss and didn't have any hesitation to bowl first.
Chris Ashton, Underhill and Moon in England team to face All Blacks
Replacements: Dane Coles, Ofa Tuungafasi, Nepo Laulala, Scott Barrett, Matt Todd, TJ Perenara, Richie Mo'unga, Ryan Crotty.
Dave Miller (51) top-scored for the Proteas before falling lbw to Stoinis following a smart review by Finch.
Australia's batsmen had crumbled in similar fashion to the sustained pressure of a South African attack that gave no quarter in the afternoon.
Australia have secured their first win of the summer and home fans are rallying behind all-rounder Marcus Stoinis - the man they are trumpeting as their saviour.
The returning Shaun Marsh (22 from 26 balls) also looked in touch but no batsmen converted stylish starts into a scores of substance.
The explosive Chris Lynn was in Twenty20 mode when he smacked 18 off the first four balls of a Rabada over before he pushed his luck too far on the fifth, mistiming a hook and was caught.
The procession continued as Glenn Maxwell flashed a cut off Pretorius through to de Kock, Stoinis heaved across the line to present Hendricks with a stinging chance at midwicket, and Cummins and Starc were both beaten for pace by Steyn.
But Marsh then edged a widish Rabada delivery to gloveman Quinton de Kock.
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