Australia have finished on 480 after Usman Khawaja's epic 180 and Cameron Green's debut Test century made India's bowlers toil for almost two days in the field.
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The tourists made the most of the best batting pitch of the series on Friday, forcing India to bowl 168 overs and take a third new ball in the fourth Test in Ahmedabad.
Australia had 10 overs to bowl at the hosts before stumps but India comfortably survived to finish 0-36, to trail by 444.
India captain Rohit Sharma (17 not out) and young opener Shubman Gill (18no) will resume batting on Saturday with Australia desperate for early breakthroughs.
Khawaja made the third-best score by an Australian in India, finishing only behind Dean Jones (210 in 1986) and Matthew Hayden (203 in 2001).
The 36-year-old opener's 10-hour epic came from 422 balls - the most deliveries an Australian has ever faced in India in an innings.
Australia comfortably reached the best total of the series, from either side, thanks to Khawaja and Green's 208-run fifth-wicket stand.
It ended up being Australia's best score in India since they tallied 577 in Delhi in 2008.
Green (114) played a brilliant, chanceless knock full of stunning drives to break through for a century that had been more than two years in the making.
The 23-year-old has made at least one significant contribution with the bat in every series he has played in since debuting at Adelaide Oval against India in December 2020.
Khawaja and Green's partnership was the second-best by an Australian pair in a Test in India, only falling short of Allan Border and Kim Hughes' 222-run stand in 1979.
"With Uzzy, he's an experienced head who's played Test cricket for more than 10 years now," Green said.
"He's so valuable for guys like myself and a few of the young guys in the team the way he goes about it.
"I'm trying to learn as much as I can off him, and, luckily enough, there are a lot of guys in the change rooms like that."
When Khawaja departed early in the final session on Friday, Australia could have easily crumbled.
But for the first time in the series, the visitors got valuable contributions from the tail.
Offspinners Nathan Lyon and Todd Murphy put on a crucial 70-run ninth-wicket stand to frustrate India's tired attack.
Lyon faced the most balls (96) in an innings during his career to finish on 34, while Murphy showed serious promise in the final Test of his debut tour.
Murphy had scored a combined five runs in the first three Tests of his career, but hit a first-class best of 41 in a huge confidence boost for the 22-year-old.
Australia's best total on tour before this innings was their 263 in Delhi as both teams have struggled to bat on three raging turners.
Star Indian spinner Ravichandran Ashwin bounced back from his second innings meltdown in Indore to take 6-91 in 48 overs, highlighted by the key scalp of Green.
As Australia batted better than they had all series, the tourists started the day beset with sadness after learning of the death of captain Pat Cummins' mother.
Cummins returned to Sydney to be around his family after the second Test loss in Delhi.
Australian Associated Press
