Set a target of 323, Australia reached 186 for six at lunch at the Adelaide Oval. The visitors need another four wickets to take a 1-0 lead in the series.
Starting from overnight 104 for four, the Head-Marsh partnership lasted only 7.4 overs before India forced a breakthrough with the old Kookaburra ball.
Head (14) was the first to go, with Ishant Sharma (1-35) bowling a sharp bouncer that followed the batsman and left him no room. The ball looped up to gully where Ajinkya Rahane made no mistake.
The duo had added 31 runs with the onus now on Marsh and Paine as the last recognised batting pair.
Marsh shouldered the responsibility and scored his first half-century in the fourth innings of a Test off 146 balls. It was his 10th Test half-century overall.
He had looked comfortable at the crease all morning, but Jasprit Bumrah (1-23) removed him after the drinks' break. The big moment came as the ball moved away just a tad and Marsh gave the slightest of edges to be caught behind in the 73rd over.
It was Rishabh Pant's ninth dismissal in the Test, equalling MS Dhoni (9 versus Australia, Melboure in 2014) as the second-best haul by an Indian wicket keeper in overseas Tests.
Cummins then helped Paine play out 10.5 overs, although he had a couple hairy moments in the 74th over off R Ashwin (2-71). India wasted a DRS review when they thought he had edged it. Four balls later, a loud appeal for caught at short leg was turned down with Cummins reviewing it successfully this time.
Rohit Sharma went off the field an hour before lunch with a lower back issue and is being receiving medical attention for the same.
India scored 250 in their first innings with Cheteshwar Pujara anchoring with his 16th Test century. Australia replied with 235 and conceded a 15-run lead. The visitors then finished 307 in their second innings, including a collapse of 5 for 25, and set a 323-run target on day four.