Australia lost a wicket in the first over of their reply and were 3-1 at stumps at St George's Park with Travis Head unbeaten on one and nightwatchman Scott Boland on two.
Bavuma produced an impressive knock, full of gritty determination but also some flourishing stroke play, as South Africa moved from a precarious 110-6 in the first session to finish the day with the upper hand.
It was the diminutive skipper’s highest test score and he also featured in two record partnerships, first with Marco Jansen and then Simon Harmer.
With Jansen, he compiled 132 runs to break the previous South Africa record of 89 runs for the seventh wicket in a test at St George’s Park, set against England in 1896, and with Harmer the highest eight-wicket partnership of 183 for South Africa in any test.
Jansen scored 85, Harmer hit his test best of 82 while Boland (5-81) registered the third five-wicket haul of his career and first outside Australia.
Bavuma’s century came up in 226 balls in a patient display while Jansen went about punishing the bowling to see the home side, who lost last week’s first test in Durban by 214 runs, lift themselves out of a potential quagmire.
South Africa resumed on 97-5 after only 43 overs were bowled on the opening two rain-affected days, but sunny conditions and a flat pitch on Tuesday allowed the batsmen to get stuck in and look to make up for lost time.
But they still looked in trouble when Wiaan Mulder was dismissed for 13, ducking into a short-pitched ball from Mitchell Starc and gloving it behind to slip Matt Renshaw. It was Starc’s 450th test wicket but was only given out after a review.
Jansen was dismissed in the first over after the second new ball was taken. He was hit in the lower midriff by Boland with the first ball and four balls later, after treatment for injury but still rattled, chipped a catch to Head at midwicket.
Harmer then joined Bavuma to continue amassing the runs before he was bowled by part-time left-arm spinner Marnus Labuschagne as he chased a maiden test ton in the afternoon gloom.
Bavuma then nicked Starc to wicketkeeper Alex Carey with the next ball to have his innings ended and promptly declared to put Australia in to bat with a handful of overs left in the day.
It took South Africa’s lead bowler Kagiso Rabada four balls on his return after a five-month layoff with a hamstring injury to have Australia opener Nic Maddinson caught at third slip by Harmer as he attempted a drive off a fullish-length ball.
He was only allowed to bowl one over, however, because of the deteriorating light, and after three more overs of spin, stumps were called.