Spinner Ravichandran Ashwin and paceman Mohammed Shami combined to tear through Australia's top order and bring India within six wickets of victory on day four of the series-opening Test on Sunday.
Shaun Marsh (31 not out) and Travis Head (11 not out) were left clinging to their wickets after nudging Australia to 104 for four at stumps, after a nerve-jangling final hour at Adelaide Oval.
Australia were 219 runs short of their victory target of 323 and will need to defy history to chase down the total.
The highest successful run chase at Adelaide Oval was in 1902, when the hosts scored 315 for six to beat England.
"We still believe we can win this," said Australia spinner Nathan Lyon who took 6-122 to help restrict India to 307 in their second innings.
"We just need to go out there and enjoy ourselves and really compete hard and fight our backsides off."
Since the second World War, no team has bettered West Indies' successful fourth innings chase of 239 for five, which defeated Australia by five wickets in 1982 at the venue.
India have never won a series-opening Test in Australia but Ashwin and Shami set up the platform to break new ground.
India are in the box seat but also rued blowing a chance to bat Australia right out of the game due to a quick and shambolic collapse after lunch.
Having resumed on 260 for five in complete control, the tourists lost their last four wickets for four runs in a panicky bout of tail-end slogging.
SCORES IN BRIEF
INDIA: First innings 250
AUSTRALIA: First innings 235
INDIA: Second innings 307 all out (Rahul 44, Kohli 34, Pujara 71, Rahane 70, Pant 28; Starc 3-40, Lyon 6-122)
AUSTRALIA: Second innings 104 for 4 (Harris 26, Marsh 31 not out; Ashwin 2-44, Shami 2-15)