Spinner Ashwin had debutant opener Marcus Harris caught for 26, breaking a 45-run partnership with Usman Khawaja, after paceman Ishant bowled Aaron Finch for a duck with the third delivery of the innings.
Usman Khawaja was 21 not out, with nervous number four Shaun Marsh on one run at the break on another steamy day at Adelaide Oval.
Virat Kohli's team were unable to add to their overnight total of 250, with tail-ender Mohammed Shami giving up the last wicket for six, caught behind after gloving a wayward leg-side delivery to Tim Paine.
Finch's place at the top of Australia's order has attracted plenty of scrutiny and the one-day specialist's home test debut got off to a horror start.
Having survived a big appeal for lbw after declining to play a shot with the second ball, he drove with force at the third delivery which nipped back and sent two of his stumps flying.
Victoria batsman Harris and Khawaja saw off India's seamers with judicious shot-making in the first hour but the former ran into trouble when Ashwin was introduced after the drinks break.
Half an hour before lunch, Ashwin very nearly had his man, as he coaxed a nick from Harris which fell inches short of Murali Vijay's grasp at silly mid-off.
Harris collected himself, smashing a lofted drive over the spinner's head two balls later and taking another four off him with a stylish cover drive.
He was undone trying to defend a fuller delivery, though, with a nick pinging off his pad and popping up obligingly for Vijay in close. (Editing by Nick Mulvenney)
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