Cheteshwar Pujara (202) and Wriddhiman Saha (117) shared a 199-run stand for the seventh wicket to propel India past Australia's first innings total of 451 all out.
Cheteshwar Pujara and Wriddhiman Saha scored hundreds to gain a formidable lead against Australia.
Unfashionable cricketers both, Pujara's and Saha's places in Team India had been questioned over the past year, but this has been a match in which substance had trumped appearance.
Saha, meanwhile, brought up his third Test ton off 214 deliveries, before the pair were dismissed - both caught by Glenn Maxwell in pursuit of quick runs to boost the Indian total ahead of declaration.
And that too by picking up Smith (21, 68b, 2x4), Australia's best batsman by a country mile and the centurion from the first innings.
Then there's Peter Handscomb who went into this series with a Bradmanesque Test average of 99.75 on the back of two centuries and two half-centuries in seven visits, three of them not out.
Before bamboozling the Australian batsmen with his left-arm spin, Jadeja also scored an unbeaten 54.
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India still trail Australia by 148 runs with six first innings wickets in hand. Pujara and Vijay however carried on to take India past the 100 run mark and keep the initiative with the hosts.
Next up will be Shaun Marsh, and how often have we said it, his Test career is again on the line.
First innings lead: 152 runs. Pujara told Hazlewood to look at the score when he was batting on 180 and when Hazlewood sledged (Saha), he told him to just go and bowl.
Cummins prised out key wickets of Kohli and Ajinkya Rahane (14) en route to his 3/49 as Indians threw their wickets cheaply on a flat track of JSCA International Stadium Complex hosting its maiden Test.
Administrators give lip service to the primacy of Test cricket, but Pujara's story - especially when contrasted with those who have barely played anything more than tennis-ball tournaments - is proof that that is just nonsense.
2001 The last time a bowler bowled 75 or more overs in an innings - Ray Price against South Africa in Bulawayo. Nathan Lyon delivered 46 overs and was clearly bothered by the cracked callus on his spinning finger.
Earlier, India started with Jadeja as he held up one end, while there was pace attack from the pavillion end but the batsmen were at their watchful best as they negotiated the first hour without any trouble.