Opener Finch cracked 109 off 127 balls and Smith hit a swashbuckling unbeaten 73 off 55 balls as Australia amassed a formidable 329 for five 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 four for 32 off eight overs, while fellow opening bowler Josh Hazlewood claimed three for 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," Smith said.
"Aaron 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 is bowling outstanding and it's great to see Mitchell Starc back here as well and getting some wickets."
Finch was in blistering form as the Australians made full use of the slow Manuka Oval pitch after winning the toss.
Finch hit nine fours and three sixes to spearhead the Australian run plunder before he was dismissed by part-time bowler de Villiers in the 41st over.
Fellow opener David Warner pounded 53 off 50 balls, with six fours and two sixes, in a first-wicket stand of 118 in 20 overs.
Warner looked set for a big score before he found de Villiers, who took a leaping catch at mid-wicket.
Smith crashed eight fours in a majestic knock, while Shane Watson took a more controlled 40 off 38 balls before he miscued and was caught in the deep by David Miller off the medium pacers of Farhaan Behardien.
Watson holds the highest ODI score at Manuka Oval with 122 against the West Indies in 2013.
Morne Morkel was the best of the Proteas bowlers with two for 84, while Vernon Philander, de Villiers and Behardien took a wicket each.
South Africa's hopes were pinned on the fourth-wicket stand of 76 between Amla and de Villiers, but once the skipper fell leg before wicket to Kane Richardson in the 38th over, the tourists were unable to keep pace with the Australian total.
Amla followed five balls later when he was bowled as he tried to slog Hazlewood through the leg-side.
David Miller looked unlucky to be given out leg before wicket to Starc on three when replays showed the ball was sliding down leg. But the team had no more reviews left to challenge the decision.