And Rohit, along with Pujara, steadied the ship for a while, putting on 45 runs.
The decision to drop Marsh leaves Australia without a fifth bowling option and gives Indian coach Ravi Shastri an obvious first line in his final team address.
Cummins provided the breakthrough of Ashwin just before the new ball.
The pace trio of Starc (2/63), Pat Cummins (2/49) and Jose Hazlewood (2/52) shared six wickets, while spinner Nathan Lyon accounted for another two as India limped their way on a good batting surface.
Pujara said it was one of his best test innings, but lamented the lack of application from India's main batsmen. All Australian bowlers apart from Travis Head claimed two wickets.
"Just little changes and little cues I used to have that I probably went away from or unknowingly went away from".
"It was a great catch and hopefully he's going to have a great couple of days with the bat as well", said the pacer. The offspinner took the only two wickets in the middle session but could not take wickets after tea.
"For the most part we were exceptional with the ball today", said Starc. The odd ball is skidding on and the odd ball is holding a bit more from the grass.
From Australia's side, Mitchell Starc started the proceedings while from the Indian side Murali Vijay and KL Rahul opened the innings. The ball went flying towards gully where Usman Khawaja dove to his left, taking a stunner to dismiss Kohli for three runs off 16 balls, reducing India to 19/3 in 11 overs.
Australia were plunged in crisis after the ball-tampering scandal in March this year saw Cricket Australia handing out year-long bans on Steve Smith and David Warner.
The dismissal of Kohli, who has scored five centuries in his two previous Test tours of Australia and rates Adelaide his favourite overseas ground, sparked wild celebrations.
Ajinkya Rahane (13) handed Hazlewood his second wicket with a rank slash outside off that ended up in the hand of the recalled Peter Handscomb, finishing off a lucrative first session for Australia. They slid further when Rohit Sharma holed out attempting a slog off Lyon.
"So I don't think spin bowling is a big threat, but it seems to me that Australia are anxious about the spinning options of India, deciding to go with Handscomb who's such a good player of spin".
A crowd of 23,802 braved the heat but the figure was considerably less than opening days for the Adelaide day-night tests of the past three years.
Earlier, skipper Virat Kohli won the toss and elected to bat first.
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