Australia were 92/5 at the tea break in their second innings, leading the hosts by 60 runs.
Glenn Maxwell (37 off 48 balls) and Peter Handscomb (18) showed positive intent with a 56-run partnership for the fourth wicket before their downfall put the hosts firmly on course for a series clinching win.
India scored 332 in their first innings courtesy Ravindra Jadeja's 63, taking a 32-run first innings lead in the process.
Bhuvneshwar Kumar, who till the post-lunch session had a horrible Test match, for once got lucky when Smith tried to pull a short delivery only to get a bottom edge that knocked back his off-stump.
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It was Umesh Yadav (2/22 in 6 overs), India's best fast bowler in recent times, who looked menacing on a bouncy pitch troubling both the openers.
After a few bouncers, Umesh cleverly pitched it up to get enough away movement to kiss David Warner's (6) bat on way to the safe hands of keeper Wriddhiman Saha.
Renshaw (8) survived a drop at third slip when the thickish edge off a Bhuvneshwar delivery was grassed by Karun Nair standing at third slip.
Maxwell however was in a mood to counter-attack as he hit debutant Kuldeep Yadav for a six over long-on. He scored runs quickly, smashing five fours and a six in the process.
Handscomb played an even better shot - inside out cover drive off Kuldeep but Ashwin got him with one that bounced sharply and the edge flew to skipper Ajinkya Rahane, who again showed his brilliance in the slip cordon.
Shaun Marsh (1) became Ravindra Jadeja's first victim as he got one to land on the crack and it jumped to hit batsman's gloves to Cheteshwar Pujara at forward short leg.