The hosts, 2-0 victors at home to England and in India in their last two series, were up against it in pursuit of 157 for nine this time – until Miller took over with an unbeaten 53 from 35 balls.
The left-hander turned the momentum with two big sixes in three balls, off Andrew Tye in the 16th over – and he duly put South Africa 1-0 up, with three wickets and four balls to spare.
Australia, who chose to bat, threatened a much bigger total – especially when opener Aaron Finch was clubbing the first three balls of JP Duminy’s over for successive sixes in a powerplay which finished with the tourists blazing away on 69 for one.
It was Imran Tahir (three for 21) and David Wiese who then did most to stall the Aussies.
The leg-spinner struck the biggest blow with his very first delivery when Finch hit a low full-toss into the hands of deep midwicket to depart for 40 off just 18 balls.
Mitch Marsh’s late hitting nonetheless ensured a testing target, and South Africa’s reply suffered an instant setback when AB de Villiers followed some Nathan Coulter-Nile outswing to be caught-behind off the first ball of the innings.
Coulter-Nile (three for 29) also got rid of South Africa’s other dangerous opener Quinton de Kock, caught and bowled playing early across a cutter.
Captain Faf du Plessis tried to get his team back on track, but his was the second of two wickets in as many deliveries after he called Miller through for a single off the mark and could not beat debutant wicketkeeper Peter Nevill’s throw.
South Africa’s long line-up of big hitters still kept them in the contest, however, and Miller bailed them out.
