Brief scores: Australia (231) beat South Africa (224/9) by 7 runs to level 3-match series 1-1.
The upshot was that South Africa were four down inside 15 overs in their chase, but with Miller and Faf du Plessis at the crease they steadied the chase.
"We weren't really happy with how we batted", Stoinis said.
That soon became 8 for 187 when Stoinis trapped Miller in front for 51, a review overturning Umpire Michael Gough's original not out decision. With the stupendous triumph, they kept their hopes alive in the series with one match remaining to be played at Bellerive Oval on Sunday.More news: The untold truth of Jim Acosta
Australia's innings slowed as Imran Tahir tied Lynn and Carey down with a tight, wicket-to-wicket line, and with singles hard to come by, Lynn banked on his boundary-hitting ability to keep the score ticking over.
Finch built a composed 41 before he was deceived by a Pretorius delivery and chopped the ball onto his wicket.
The fast bowler needed the intervention of one of South Africa's top lawyers - Dali Mpofu - to escape‚ on appeal‚ a ban for a shoulder charge on Steve Smith after he dismissed the then Australia captain during the St George's Park Test in March.
Their form slump since has been characterised by batting collapses and they again struggled to compile a big total with some tight South African bowling putting them back in the pavilion in 48.3 overs.
But, along with Shaun Marsh (22 from 26 balls), the top-order batsmen failed to fully cash in.More news: Chris Ashton, Underhill and Moon in England team to face All Blacks
Proteas paceman Rabada picked up 4-54, Dwaine Pretorius took 3-32 and Dale Steyn bowled superbly for his 2-31 from 10 overs.
Lynn then seemed to take control of the innings, and in the 27th over got stuck into Kagiso Rabada, hitting him for a six and then three consecutive fours.
Finch had stressed beforehand the need to go back to basics and build partnerships, and he and Travis Head tried to set the tone after being sent into bat.
Finch put on 30 runs with Lynn before the skipper departed, chopping a short ball from Pretorius onto his stumps. Pacer Peter Siddle, who was named as cover for Starc in the recent overseas T20I series against Pakistan, has also been dropped while the other frontline quicks - Josh Hazlewood and Pat Cummins - continue to be left out too, despite being fit, according to an International Cricket Council (ICC) release.
Aiden Markram and Reeza Hendricks looked unsafe as they pushed the score to 46, but they needlessly attempted a third run from a clip off the pads and Markham was run out.More news: Priyanka Chopra, Nick Jonas reportedly obtained license for marriage