Looking to wrap up the four-match series 3-0, the hosts dismissed India for 475 shortly before tea to take a 97-run first innings lead and then embarked on a fierce assault on the Indian bowlers in a bid to force a result.
It started poorly when spinner Ravichandran Ashwin (4-104)had David Warner (4) caught in the slips after facing five balls and Shane Watson played on for 16.
Opener Chris Rogers, though, hit a 77-ball 56 before he was caught at midwicket off Bhuvaneshwar Kumar and Smith was soon back in his considerable stride.
The Australian captain raced to his half century in 44 balls and had soon passed Donald Bradman as his country's highest scorer in a series against India.
Shaun Marsh came and went for a single run to give Ashwin his third victim before Smith's 93-minute innings came to an end when Mohammed Shami trapped him plumb lbw.
His 71 brought his series tally to 769 and a group of fans in the member's stand bowed low to him as he walked back to the Sydney Cricket Ground dressing room to ponder his declaration.
Joe Burns took up the mantel and grabbed his maiden Test half century with three sixes and eight fours in his 39-ball 66, while Brad Haddin ably assisted with 31 not out.
India captain Virat Kohli has been no slouch with the bat in this series but he was only able to add seven runs to his overnight score before clipping a Ryan Harris delivery to Rogers at midwicket some 20 minutes into play.
The 147 helped bring his tally over the four Tests to 646 runs, the second highest by an Indian batsman in an overseas series after the 774 Sunil Gavaskar accumulated in West Indies in 1971.
Despite the Indian tail wagging for the first time in the series and good batting conditions, the Australians managed to winkled out four more batsman.
They got some help from the third umpire to get rid of Kumar (30), who put on 65 with Ashwin (50) for the eighth wicket, after a referred decision that will do little to persuade India to embrace the Decision Review System.
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