In reply to India's first innings of 250, Australia reached lunch at 57 for two with Usman Khawaja unbeaten on 21 runs and Shaun Marsh one not out. Finch was dismissed in the first over and Marcus Harris, making his test debut, fell just before lunch.
After a scorching opening day, conditions remained warm but cloud cover provided favorable conditions for India's seamers. Australia's innings started disastrously when Finch was bowled by Ishant Sharma.
The hard-hitting Finch drove loosely at a full delivery and inside-edged onto his stumps, which went cartwheeling. Finch became just the fifth Australian batsman to be dismissed without scoring in their home test debut.
Harris scored immediately with a crisp three runs on the leg-side. He found the going tougher after that, being unable to score a run off his next 15 deliveries. Harris showed discipline and concentration in a notable contrast to India's reckless batting approach on the first day.
India spinner Ravichandran Ashwin came on after the drinks break and immediately caused problems for the left-handed Harris. Fifteen minutes before lunch, Harris prodded a full Ashwin delivery to Murali Vijay at silly point. Harris made 26 runs from 57 balls.
India had a scare when paceman Mohammed Shami required medical attention on his right shoulder with one ball left in his fourth over. He left the ground but returned shortly after.
Australia dismissed India on the first ball of the morning, with Shami edging a short Josh Hazlewood delivery to be caught behind.
India had sustained a blow late on the first day when Cheteshwar Pujara was run out after compiling 123 runs. No. 6 batsman Rohit Sharma was next best with 37 runs as India's batting struggled on a two-paced Adelaide Oval pitch.
Hazlewood finished with 3 for 52 from 20 overs, as wickets were shared among Australia's four-pronged attack.