An aggressive Marcus Harris played a loose cut stroke, edging a catch behind off Shami. Harris made 26 in both innings of his test debut.
Australia were teetering at 60-3 when Ashwin claimed the prized target of Usman Khawaja, who threw away his wicket in holing out to a forward-diving Rohit Sharma for 8.
Peter Handscomb (14) combined with Marsh to temporarily blunt India before he lost patience, spooning a catch to midwicket off Shami.
Spinner Nathan Lyon said the team would find inspiration from their recent drawn test against Pakistan in Dubai. Australia batted out 139.5 overs to finish on 362-8.
"We spoke about Dubai. We still believe we can win this," Lyon said. "I think the pitch has quickened up. The wicket is good enough for us to have the belief to score these runs."
India suffered a scare in the final hour when paceman Jasprit Bumrah left the ground after falling awkwardly in the field. He received treatment on his right shoulder and returned to bowl.
Australia's batting effort soured their earlier comeback after India collapsed in their second innings, losing seven wickets for 73 runs. The visitors' aggressive approach after lunch backfired, with four wickets falling for only four runs. Half-century maker Ajinkya Rahane added just 13 in the second session, falling for 70.
Lyon claimed 6-122 from 42 overs and paceman Mitchell Starc shrugged off a disappointing first session to finish with three wickets.
Rahane and Cheteshwar Pujara shared an 87-run fourth-wicket partnership, the highest of the match. Pujara made 71 off 204 balls, following on in fine style from his first-innings century.
India assistant coach Sanjay Bangar praised Pujara's efforts.
"A big score in Australia has been elusive for him and he would be satisfied to have played a role for his team," he said.
India have not won a test in Adelaide since 2003, when they beat a Steve Waugh-led Australia by four wickets.
- AP
