Mohammed Siraj snared five wickets as Australia was dismissed for 294 late on day four, setting India a target of 328 to win the series-deciding cricket Test at the Gabba.
The highest successful run chase at the Brisbane venue was in 1951, when Australia finished 236-7 against the West Indies. The Australians haven't lost at the Gabba since 1988, but nobody is counting India out.
The openers faced 1.5 overs before rain stopped play for a second time on Monday and India was four without loss - one scoring stroke from Rohit Sharma. Stumps was called shortly after 5 pm local time, leaving India one day to try to get 324 more runs.
Siraj put down two catches before picking up the key wicket of Steve Smith for 55 in the middle session as India's injury-depleted bowling attack kept taking the game to Australia.
Australia was 243-7 at tea on day four - a rain shower forcing an early break - with an overall lead of 276 and a declaration seemingly imminent.
But the hosts continued batting when play resumed in gloomy conditions in the evening session, adding a further 51 runs for three wickets.
Pat Cummins finished unbeaten on 28. Nathan Lyon, in his 100th Test, took Australia's lead past 300 when he hooked a six off Siraj but was out soon after for 13, giving Thakur his fourth wicket of the innings and seventh of the match.
Thakur took the catch off Siraj's bowling to dismiss last man Josh Hazlewood (9). Siraj, leading the India attack in just his third Test in the injury-enforced absence of Jasprit Bumrah and spinner Ravichandran Ashwin, returned 5-73 for his career-best figures.
Smith said the match was well-poised, and typical of a series that as "ebbed and flowed" throughout. Australia won the first Test by eight wickets in Adelaide, India responded with an eight-wicket win in Melbourne and then get the emotional better for a draw in the third Test in Sydney, where the tourists batted through the last four sessions to salvage a draw.
"The game is in a nice place for us," Smith said. "The wicket started to play a few tricks today, a couple of balls shot up, so tomorrow is going to be (about) letting the natural variation of a day five wicket do its work, and hopefully we can hold onto our chances."
Smith said there was a lot more happening with the wicket at the Gabba and it was different to the Sydney Cricket Ground, where India batted for 130 overs to save the match.
"For us (the key) is being patient, not searching too much. Letting it happen," Smith said. "The guys are excited about tomorrow."
He tipped Lyon, three shy of 400 Test wickets, to play a big role with some cracks on a day-five pitch to aim at. He also expected allrounder Cameron Green to use his height and the unpredictable bounce to get his first wicket in Test cricket.
Australia started play 21 without loss on Monday and cruised to 89-0 on the fourth morning before losing four wickets in six overs to slip to 123-4.
Smith then combined with Cameron Green to lift the total to 196 before a sharply-rising ball from Siraj hit him on the glove and deflected away to Ajinkya Rahane in the gully.
Smith, who got a reprieve on 42 when he advanced to off-spinner Washington Sundar and Siraj put down a chance in the deep, reviewed the decision, thinking his hand wasn't on the bat at the time. But replays clearly confirmed the dismissal.
Green (37) had a reprieve when Siraj put down a sharp return catch, but his 90-ball innings ended when he edged Thakur to Rohit Sharma at second slip to make Australia 227-6. Australia captain Tim Paine scored a quick 27 before he was caught behind attempting a hook off Thakur just before the tea interval.
Thakur had earlier picked up the first wicket of the innings when Marcus Harris (38) gloved a catch to wicketkeeper Rishabh Pant in the 25th over.
David Warner compiled 48 from 75 deliveries before he was out in the following over, trapped lbw by Sundar to make it 91-2.
Siraj chimed in with two wickets in an over to remove Marnus Labuschagne, who scored 25 from 22 balls, and Matthew Wade (0) in the same over.
While the series is level at 1-1, India needs only a draw to ensure it retains the Border-Gavaskar trophy.