The Proteas were bowled out for 408 after lunch, a first-innings lead of 163.
Elgar took his overnight score of 140 to 185 before he gloved a catch down the leg side off Shardul Thakur.
Elgar and Jansen (84*) had batted for two hours in a morning’s play extended to two-and-a-half hours when Elgar fell after a 287-ball innings which included 28 fours.
The pair took South Africa from a narrow 11-run lead at the start to a position of strength.
Elgar and Jansen put on 111 for the sixth wicket.
Having survived a tricky period in fading light before the close on Wednesday, they twice saw off the threat of Jasprit Bumrah and Mohammed Siraj – at the start of play and later with the second new ball.
The tall Jansen made his second Test half-century and highest score.
He faced 147 balls and hit 11 fours and a six.
Proteas captain Temba Bavuma did not bat due to a hamstring injury.
© Agence France-Presse
Photo: Sydney Seshibedi/Gallo Images
