Elgar built key stands, including an unbeaten 114 runs with Quinton de Kock, on 69, to thwart the Indian bowling threat in Visakhapatnam.
Elgar and de Kock still need 11 runs to avoid the follow on as they chase India's 502 for seven declared.
The left-handed Elgar, who started the day on 27, reached his hundred with a six off Ravichandran Ashwin in the second session.
He also put on 115 runs with skipper Faf du Plessis for the fifth wicket to revive the innings after South Africa lost crucial wickets late Thursday.
Ashwin broke the stand, getting du Plessis caught at leg gully for 55 against the run of play.
But Elgar stayed firm with positive strokeplay as he hit 14 fours and four sixes.
South Africa started the day on 39-3 but fast bowler Ishant Sharma struck early, taking overnight batsman Temba Bavuma lbw for 18.
Left-arm spinner Ravindra Jadeja also posed problems and nearly got Elgar on 74 but wicketkeeper Wriddhiman Saha spilt the chance.
Indian opener Mayank Agarwal set the tone for the hosts' dominance of the game with his 215 and a 317-run opening stand with Rohit Sharma, who hit 176 of India's giant score.
An over-enthusiastic fan caused a brief security scare as he rushed on the field to get a selfie with the India captain Virat Kohli.
Kohli, who is treated as a national hero, was taken by surprise by the intrusion in between overs. The youth tried to hug him before taking a picture and touching the star player's feet.
Kohli tried to order the teenager off and vice-captain Ajinkya Rahane also intervened but the persistent fan stayed on the field for more than two minutes. Security officers eventually overpowered him as he ran off, while the crowd roared and clapped.