Australia removed the dangerous Rishabh Pant and Cheteshwar Pujara before tea on day five Monday to leave them needing five wickets to win the third Test and India battling for 127 more runs to pull off an unlikely upset.
India's hopes in the final session rest on the shoulders of an injured Hanuma Vihari, whose movement is hindered after tweaking a hamstring, and their tailenders.
Vihari was not out four and Ravi Ashwin on seven as they reached 280 for five.
Australia grabbed the prize wicket of Pant for 97 as his explosive innings fell short of a century when he was caught by Pat Cummins off Nathan Lyon going for another big hit.
Josh Hazlewood bowled the stoic Pujara, who brought up 6,000 Test runs before his resistance was broken after a battling 77 off 205 balls.
India resumed at 98-2, still needing a further 309 runs after being set a huge 407 to win after Australia declared their second innings at tea Sunday on 312 for six.
No team has ever bettered 288 to clinch victory in the fourth innings at the Sydney Cricket Ground, with Australia achieving that mark against South Africa in 2006.
Australia accounted for openers Shubman Gill (31) and Rohit Sharma (52) before stumps on day four, and when Lyon got captain Ajinkya Rahane in Monday's second over without adding to his overnight four the end appeared nigh.
But Pant, who took a nasty blow to his elbow while batting in the first innings and didn't keep wicket on Sunday, surprisingly came in at number five ahead of Vihari and produced some fireworks.
He had no strapping on the hurt elbow and, after settling in and surviving a missed catch by Tim Paine on three, began to hit out, smacking a four and a six off Lyon on consecutive balls.
Batting more like it was a Twenty20 match, he then clobbered two more sixes in a row off Lyon before bringing up his third Test 50 off just 64 balls.
Paine dropped him again on 56, both times off Lyon.
He kept the scoreboard ticking over after lunch and a third Test century looked imminent only for the 23-year-old to throw his bat one time too many.
His wicket shattered a 148-run partnership with Pujara, who resumed the day on nine and slowly reached a second consecutive 50.
In the process he brought up 6,000 career runs, the 11th Indian to achieve the feat, before falling to Hazlewood.
Vihari hurt his hamstring on a quick single, adding to India's injury list. It is not clear whether Ravindra Jadeja, who has a fractured thumb, will pad up if needed after tea.
India were bowled out for 244 in their first innings in reply to Australia's 338.