South Africa's pace bowlers plowed through the patched-up Sri Lankan order in just 40.3 overs at the Wanderers after Sri Lanka chose to bat first, with Anrich Nortje taking a career-best 6-56.
Du Plessis was on 55 and Bavuma on 41 as stumps were called on Day 2 in Centurion.
Du Plessis had resigned as Test captain in February and Quinton de Kock was expected to take the reins but Cricket South Africa ruled him out of the job, saying they don't want to overburden the wicketkeeper-batsman.
Elgar believes head coach Mark Boucher leans on him a lot and that they both share a great understanding.
With the early dismissal, Elgar became only the fourth South African batsman to be dismissed on the first ball of a Test match after Gary Kirsten fell before West Indies pacer Curtly Ambrose in Cape Town.
Elgar was hit just above the ear and immediately fell on the ground, calling for medical help. He was batting on 16 off 29 balls when Umesh bouncer struck him.
Earlier on Day 3, Elgar was hit by a nasty bouncer by Indian pacer Umesh Yadav on the helmet in the 10th over of second innings before Tea.
Dean Elgar was retired hurt after he was struck on the helmet during the third day of the final Test.
After managing a 1-1 draw in the T20 series, South Africa suffered big defeats in the first two Tests of the three-match rubber against a dominant outfit led by Virat Kohli.
Dean Elgar was among the South African batsmen who had a disastrous tour of India four years ago.
Speaking at the end of the day's play, Elgar said that it is really tough to bat in Indian conditions and he learnt from the experience he had on his last tour with the team.
Dean Elgar and Quinton de Kock kept the skilful Indian spinners at bay with memorable hundreds, orchestrating South Africa's spirited counter-offensive against the fancied hosts in the series-opening Test.
The Proteas dominated the first two sessions of the day but the last session was clearly shared as the spin duo of Ashwin and Ravindra Jadeja clawed back.
Dean Elgar century was South Africa's first Test century in India since Hashim Amla's triple-figure knock in 2010. But for Elgar, its significance lies elsewhere.
On Thursday, Steyn bowled a fiery unbroken 10-over spell with speeds constantly in the high 140s.
Faf Du Plessis was suspended for one Test and fined 20 percent of his match fee after his team were found guilty of maintaining a slow over-rate during South Africa's 9 wicket victory in the second Test.
Arthur was found to have violated Article 2.8 of the ICC Code of Conduct which relates to "showing dissent at an umpire's decision during an international match".
Faf and Sarfraz failed to open their accounts with the bat in both innings.
South Africa ground their way toward a series win with an unbeaten 40-run partnership between Dean Elgar (39 not out) and skipper Faf du Plessis (34 not out).
Cummins removed AB de Villiers for 64 after tea to end his 128-run stand with Dean Elgar and start South Africa's sudden slide.
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