Bavuma’s career-best 179, made from 334 balls in 381 minutes, was the cornerstone of the recovery, while Simon Harmer’s 82 provided the other half of a decisive eighth-wicket stand that eventually reached 183.
SA had resumed on 97/5 after losing five wickets in the first two days, with Bavuma unbeaten on 41 and Wiaan Mulder on two after the second day had been reduced by rain.
The pressure remained intense when Mulder was dismissed for 13 by Mitchell Starc at 110/6, leaving Australia in a position to press home their advantage.
Jansen, however, changed the momentum with an aggressive innings that gave Bavuma the support he needed.
The righthand batter attacked the Australian quicks, reaching his half-century from just 43 balls as the pair began to wrestle control away from the visitors.
Their seventh-wicket partnership produced 132 runs from 166 balls in 139 minutes, with Jansen eventually falling for 85 off 100 balls.
His innings included nine fours and three sixes, and it gave SA the impetus they had been missing after Australia’s attack had reduced them to 110/6.
Bavuma continued to absorb the pressure at the other end, passing his half-century off 140 balls before reaching his fifth Test century from 226 deliveries.
The captain’s hundred contained 11 fours and a six, but his innings eventually grew into something much bigger as he surpassed his previous Test best of 172.
Jansen’s dismissal at 242/7 again gave Australia an opportunity to finish the innings, but Harmer joined Bavuma and the pair settled into another decisive partnership.
The eighth-wicket stand grew to 76 by tea, with SA on 318/7 after 101 overs. Bavuma was on 125 and Harmer on 30.
They continued to frustrate Australia after the interval, first taking the partnership beyond 100 before pushing SA to the 400-mark.
