Marnus Labuschagne made 91 off 196 balls, and shared a century stand with Smith to see Australia past 200 runs for the first time in five innings this series, before he edged the impressive Jadeja to Ajinkya Rahane at slip with Australia at 206-3.
Jadeja then had Matthew Wade (13) playing a poor shot to be caught in the deep, before Jasprit Bumrah (2-66) trapped Cameron Green lbw with the last ball before lunch.
After lunch, captain Tim Paine (1) and Pat Cummins (0) were both bowled cheaply leaving Australia at 278-7.
Smith then reached his 27th hundred in 76 tests with a leg glance for three runs off Navdeep Saini in the 98th over. The former captain was far more effusive in his celebration than usual reflecting the relief at having scored his first century in 15 innings spanning four series.
Smiths eighth 100 against India came in 259 minutes off 201 balls, with 13 boundaries, and was the first against India in a home test since David Warners quickfire century at the SCG in 2015.
It was also Smiths first century in Australia since December 2017 in a drawn test against England at the Melbourne Cricket Ground.
Starc scored 24 off 30 balls, including two boundaries and a six, before holing out from debutant Saini (2-65).
Nathan Lyon was also out for a duck, trapped by Jadeja to reduce the hosts to 315-9.
Smiths was the last wicket to fall, run out spectacularly by Jadeja with a direct hit from deep square leg, as he was chasing quick runs with Josh Hazelwood.
India opened its innings briskly with second-gamer Gill especially looking comfortable at the crease.
Hazlewood made the lone breakthrough for the hosts in the 27th over to remove Sharma caught and bowled for 26, leaving India at 70-1.
Gill reached his maiden half century off 100 balls, with eight boundaries, before Cummins found the openers edge and had Gill caught by Green at gully the next ball.
Pujara and Rahane both survived some near misses in the final overs as Australia bowled a perilous line and length searching for a late wicket to tilt the match in their favor going into the third day.
On Thursday, Will Pucovski set the tone with a flourishing 62 on debut, before Labuschagne and Smith took over as Australias batting lineup finally clicked against India on a rain-interrupted opening day.
The four-match Border-Gavaskar trophy series is level at 1-1. The holder India is hoping to secure it with a victory in Sydney and Australia needing to win the series to reclaim it.
Australia won the opening day-night match in Adelaide by eight wickets and India responded with an eight-wicket win in Melbourne last week.
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