Smith's 117 not out is only the second century in the spin-dominated series -- following his 109 in the first test in Pune -- as the 27-year-old joined the 5,000-run club in his 53rd test.
Team mate Glenn Maxwell celebrated his return to test cricket with a career-best 82 not out, his burgeoning partnership with Smith already worth 159 runs on the opening day of the contest.
Home captain Virat Kohli injured his shoulder while diving to save a boundary in the 40th over and missed the rest of the day's play.
Smith won an important toss and had little hesitation in batting first, something of a ritual on India's low-bounce tracks, and Matt Renshaw led their brisk scoring with a boundary binge.
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With an extremely fast outfield at the Jharkhand State Cricket Association (JSCA) Stadium hosting its first test match, Renshaw's first 24 runs all came via boundaries as the tourists cruised to the 50-run mark in the 10th over.
Left-arm spinner Ravindra Jadeja got the breakthrough in his first over when David Warner slapped a juicy full toss back to the bowler to depart for 19.
Renshaw looked good for his third fifty in the series before edging Umesh Yadav to slip, having played and survived an identical shot on the previous delivery. Renshaw's fluent 44 included seven elegant boundaries.
"I thought we all got starts but just didn't get on with it and make a big score like Steve did. He showed us how to go today," Renshaw said, admiring how his captain managed to return his focus to the game after the review row.
"He's handled the whole situation really well and just showed that we're here for a cricket series and showed how to do it today."
Australia were reeling at 89 for three when Smith and Peter Handscomb (19), protagonists of the review controversy in Bengaluru, joined forces for a half-century stand.
Handscomb had advised Smith to gesture towards the dressing room in Bengaluru for advice on whether to review a leg-before-wicket decision, a move that drew explosive response from Kohli resulting in considerable acrimony in the middle of the four-test series.
DISCIPLINED MAXWELL
Yadav trapped Handscomb (19) lbw with an inswinging yorker but Smith found an able partner in Maxwell, who eschewed his natural aggression, and played judiciously to justify his selection, replacing the injured Mitchell Marsh.
Known for his destructive batting which makes him a limited-overs asset, Maxwell showed tremendous self-restraint, taking 57 deliveries to hit his first boundary.
There were glimpses of his natural aggression as well, as when he hit Jadeja for the first of his two sixes to bring up his maiden test half-century.
Maxwell surpassed in his very first innings of the series the 48 runs that Mitchell Marsh had totalled in his four during the first two tests and the all-rounder is not done yet.
Australia also brought in paceman Pat Cummins, who is playing his second test after more than five years in the wilderness, to replace the injured Mitchell Starc.
India welcomed back opener Murali Vijay, who missed the tempestuous Bengaluru test with a shoulder injury.
(Editing by Sudipto Ganguly, Neville Dalton)
