The hosts posted a commanding 488 all out in their first innings as middle-order batsman Bavuma was denied his second test century when he ran out of partners.
Australia will start their innings after the tea break, but their hopes of a victory that would level the series at 2-2 look very slim, especially with rain predicted later in the test, as their controversial tour looks to be coming to a meek end.
New Australia captain Tim Paine had complained after the opening day that his side were "flat" at times and they appeared so again, the usual intensity in their performance visibly lacking.
Seamer Pat Cummins was comfortably the most effective bowler, recording excellent figures of 5-83 as he troubled the South African batsmen throughout the innings, but he was not backed up by his fellow bowlers.
The home side resumed their innings on day two on 313 for six under cloudy skies that should have made for a bowlers’ morning.
Bavuma and Quinton de Kock (39) initially showed patience, with the former adding just a single run to his overnight score of 25 in the first hour of play.
But both were more expansive after that as South Africa looked to drive home their growing advantage.
De Kock got a top-edge to a Nathan Lyon (3-182) delivery to be caught at mid-on, bringing to an end a stand of 85 for the seventh-wicket, while Vernon Philander (12) became the spinner's third victim when he holed out in the deep.
A career-best 45 from Keshav Maharaj added another 76 for the ninth wicket as South Africa’s tail wagged, but his stay was ended when he edged Cummins to wicketkeeper Paine.
Any hopes that Bavuma might get the five runs needed for his deserved tonne were ended with the very next ball when Morne Morkel edged to Peter Handscomb at second slip.
The 10th test half-century for Bavuma will still be a relief for him and South Africa, who struggled in the number six position this year before his return from a finger injury.
Australia have been rocked by the ball-tampering scandal that saw captain Steve Smith, his deputy David Warner and Cameron Bancroft sent home with heavy sanctions.
South Africa lead the series 2-1 after winning the last two tests in Port Elizabeth and Cape Town.
(Reporting By Nick Said; Editing by Amlan Chakraborty and Ken Ferris)
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