Finch's 102 came on the back of the two centuries he scored against England at the start of the year, and was supported by fifties by Phillip Hughes and George Bailey.
Finch shared an opening stand of 92 with Hughes, who was brought into the side as Australia left out spinner Nathan Lyon, and then added 114 for the third wicket with Bailey.
While it was thought that Australia would be the rustier of the two sides given that their only ODI since January was Monday's 198-run win over Zimbabwe, it was South Africa who looked as though their minds were still on the fishing break they enjoyed after beating Zimbabwe 3-0 in a series last week.
Legspinner Imran Tahir was able to stem the flow when he dismissed Hughes for 51 and then Mitchell Marsh, but Finch and Bailey responded by scoring at more than seven runs per over in their partnership.
Ryan McLaren broke the stand when he had Finch caught at deep square-leg, and Morne Morkel backed that up with the wickets of Bailey, who was caught behind for 66, and Glenn Maxwell.
However, Steve Smith clubbed 31 from just 19 deliveries and then Mitchell Johnson scored an unbeaten 23 from eight deliveries to assure Australia of a strong finish.