The tourists removed Smith for just 14 after the Aussie skipper scored 192 in the first innings in his third century of the series.
Veteran opener Rogers reached his fourth consecutive half-century of the series but again failed to go on with it and was bowled by Ravi Ashwin for 69 nearing tea.
At the break, Australia were 174 for four with Shaun Marsh on 15 and debutant Joe Burns not out eight.
Rogers was dropped on 33 in the slips by Shikhar Dhawan on the second ball after a lengthy rain delay and had another big moment on 56 before his dismissal.
Umpire Richard Kettleborough called for the video umpire to check if the ball had carried in a claimed catch by M.S. Dhoni off Mohammed Shami.
But replays showed the ball hitting the ground in Dhoni’s glove and Rogers was given a reprieve. He added another 13 runs before he edged on to his stumps.
India rejoiced with the big pyschological wicket of Smith in one of their rare successes against the prolific Aussie skipper in the series.
Smith was caught at leg-slip by Ajinkya Rahane off Umesh Yadav after playing a similar shot for a boundary a few overs before.
Shane Watson again failed to nail a big score when he was caught behind off Ishant Sharma for 17.
Australia also lost explosive opener David Warner before lunch after he was looking dangerous.
Warner hit a boundary and an all-run four off his opening two balls and threatened to quickly extend Australia’s lead before he fell in Ashwin’s fourth over.
He played forward but was struck on the back pad and was given out leg before wicket for 40 off 42 balls with six fours.
Australia quickly mopped up India’s first innings for 465 to lead by 65 with Mitchell Johnson snaring both wickets of Umesh Yadav and Mohammed Shami for the addition of just three runs to the overnight score.
Johnson had Yadav caught behind off the second ball of the day and in his next over had Shami caught by Smith at second slip for 12. Johnson finished with three for 135.
India lost their last seven wickets for 56 after Virat Kohli (169) and Rahane’s (147) ground record 262-run partnership for the fourth wicket.
It was yet again another lower order collapse to highlight one of the main differences between the two sides.
Australia’s lower order from the sixth wicket on have put on 784 runs in the series compared to India’s 389.
Australia only need a draw to lift the Border-Gavaskar Trophy after winning the opening two Tests of the four-match series in Adelaide and Brisbane. (AFP)