The hosts needed three runs from the last over to reach their target of 146 and White, despite losing partner Sean Abbott and almost his own wicket in an attempted run out, held his nerve to deliver with a rushed single.
After the side that batted second dominated the first two matches in Adelaide and Melbourne, Australia captain Aaron Finch opted to field first when he won the toss at the Olympic Stadium.
Initially it did not look like an astute move as openers Quinton de Kock (48) and Reeza Hendricks (49) put on 75 runs for the first wicket as the Proteas raced to 84 for one in their first 10 overs.
With all rounder James Faulkner (3-28) to the fore, however, the Australian bowlers upped their game in the second half of the innings and restricted South Africa to what looked like a below par 145 for six.
Finch led from the front and started the run chase at a canter, firing 33 off 25 balls before pulling a Marchant de Lange delivery straight to David Wiese at backward square leg.
Ben Dunk (14), Nic Maddinson (4) and Shane Watson (5) followed quickly and the hosts were off the pace on 76 for four at the halfway mark of their innings.
Glenn Maxwell upped the scoring rate in a 15-ball cameo for 23 runs that featured a huge six, a reverse paddle and a reverse sweep for four before a big heave to deep midwicket was caught by Hendricks.
That gave experienced left-arm spinner Robin Petersen (3-28) his third wicket but White soon picked up the pace and held down one end to steer Australia to victory.
(Reporting by Nick Mulvenney, editing by Ed Osmond)