Opener Finch cracked 109 off 127 balls and Smith hit a swashbuckling unbeaten 73 off 55 balls as Australia amassed a formidable 329-5 off their 50 overs in Canberra.
The Proteas were always up against it and were dismissed for 256 in the 45th over, with last man Imran Tahir unable to bat because of a knee he injured while fielding.
The Australians now lead the five-match series 2-1 ahead of the final games in Melbourne on Friday and Sydney on Sunday.
Left-armer Mitchell Starc troubled the South Africans with his late swing and finished with 4-32 off eight overs, while fellow opening bowler Josh Hazlewood claimed 3-51.
Hashim Amla hit 102 off 115 balls and skipper AB de Villiers cracked a belligerent 52 off 34 balls to briefly threaten Australia's bowlers.
Smith was named man-of-the-match for his innings which was full of improvised shots, including one boundary cheekily hit through his legs.
"It was a free hit, so I had pretty free rein to do what I wanted. I was lucky enough it came off and went to the boundary," said Smith.
"Finch was able to lay the foundation and I was able to close it out. The boys hit their lengths well and bowled really well, we got the ball quite soft and it started reversing.
"Josh Hazlewood’s bowling is outstanding and it's great to see Mitchell Starc back here as well and getting some wickets."
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