South Africa were 293-9 at the close of the fourth day, still 123 runs adrift of Australia, who will return on Monday to wrap up what appears an almost inevitable victory and a 1-0 lead in the four-match series.
South Africa's opener Aiden Markram provided some stout resistance, scoring 143 runs, while Quinton de Kock's unbeaten 81 also helped frustrate the visitors as South Africa recovered from a horrid start to their second innings.
Australia looked to be heading to an easy win when they set South Africa 417 for victory before taking four key home wickets for just 49 runs before lunch as they devastated the home team's top order.
Dean Elgar was the first to go for nine before Josh Hazlewood claimed the wicket of Hashim Amla -- the sixth time he has done so in the last four tests between the two countries.
AB de Villiers was run out after facing a single ball and then captain Faf du Plessis scored only four before having his stumps knocked out of the ground by Pat Cummins.
Yet the 23-year-old Markram proved a bulwark with a knock that belied his inexperience and gave the home fans a tantalising glimmer of an improbable victory.
He was the mainstay in an 87-run partnership with Theunis de Bruyn, who scored 36, and added 147 more runs with De Kock for the sixth wicket.
South Africa were 135 runs off their victory target when a tired Markram tried to guide the ball down to third man and was caught behind by Tim Paine off Mitchell Marsh's medium pace bowling.
Starc then returned to the attack, deciding to forego the second new ball in favour of swinging the old one, and had immediate success by claiming three wickets in a single over.
Vernon Philander (6) was caught behind, followed by the wickets of Keshav Maharaj and Kagiso Rabada, who were bowled with successive deliveries.
Yet Starc was denied a chance to complete his hat-trick -- and take his tally to 10 wickets in the match -- due to bad light with the umpires only allowing play to continue if Australia used spinners.
Tailender Morne Morkel held off for 27 deliveries, without scoring, to force play into the final day.
(Reporting by Mark Gleeson; Editing by Toby Davis)
