At the tea break, the Australians were 27 without loss and 100 runs away from victory with David Warner on 20 and newcomer Matt Renshaw three not out.
The South Africans thought they had taken the wicket of Renshaw in the second over when they claimed a slips catch by Hashim Amla off Vernon Philander.
But umpire Richard Kettleborough was unmoved and a review showed no contact with Renshaw's bat.
It was the ninth review in this series involving Englishman Kettleborough, who has a 100 percent success rate.
South Africa will be hoping to repeat their predecessors' famous five-run victory in Sydney 22 years ago when they bowled out Australia for 111.
The Proteas, who thrashed Australia in the first two Tests to claim a third consecutive series Down Under, added 56 runs to their overnight lead before they were dismissed for 250 with opener Stephen Cook scoring a defiant century.
Cook raised his second Test century and first against Australia with a pull through square leg for four off Josh Hazlewood.
It was a big finish to an underwhelming series for Cook, who had scores of 0, 12, 23 and 40 in his other innings before his timely ton off 235 balls.
Cook was the last man out when he was bowled by Mitchell Starc 45 minutes before the tea break.
Quinton de Kock, who mastered the Australian bowlers in the first two Tests, fell for just five.
Australia sought a review after Jackson Bird's leg before wicket appeal was turned down and replays showed the ball hitting de Kock's front pad pitching on middle and leg stump.
It was a massive blow for the Proteas' hopes of posting a defendable target for Australia to chase down in the fourth innings.
Kagiso Rabada lasted just nine balls before he edged a leg-side delivery from Hazlewood to a diving Matthew Wade.