Jaiswal and Kohli slam centuries as Australia stare at defeat
The visitors declared their second innings on 487-6 in the final session of day three after Kohli rolled back the years to reach 100.
Faced with an ominous 534 to win, Australia needed to survive 30 minutes before stumps.
But they instead imploded with rookie Nathan McSweeney out for a duck to Jasprit Bumrah and nightwatchman Pat Cummins falling for two, edging Mohammed Siraj to Kohli at slip.
When Bumrah trapped Marnus Labuschagne (3) lbw, a close was called with Usman Khawaja not-out three and Australia in disarray at 12-3.
The day belonged to Jaiswal, who grew up homeless in Mumbai, hitting 15 fours and three sixes in a mature 297-ball stay.
The 22-year-old opener, in his 15th Test, seemed destined for a double century but he cut Mitchell Marsh to Steve Smith at point and departed to a standing ovation.
"For me all my centuries are amazing, but this is special because I really wanted to do it (against Australia)," said Jaiswal.
"I worked so hard in every practice session, I wanted to score runs here in Australia, so I really enjoyed it.
"I was just playing normally, I always trust in me and believe in me."
Kohli was also on fire in a welcome resurgence after a lengthy lean spell that sparked questions about his place in the team.
The 36-year-old rode his luck to bring up a 30th Test ton and his first since July last year to silence the critics.
Along with Jaiswal, Australia removed Devdutt Padikkal (25), Rishabh Pant (1), Dhruv Jurel (1) and Washington Sundar (29), having accounted for KL Rahul (77) before lunch.
Rahul's demise snapped a 201-run opening partnership with Jaiswal, a record for India in Australia, eclipsing the previous best set by Sunil Gavaskar and Kris Srikkanth in Sydney in 1986.
"I think it's just about the batters sticking to their plans tomorrow, batting some time," said Australian speedster Josh Hazlewood on what Australia now need to do.
"It's obviously a long series so if we can put some overs into their top quicks, that's one of the goals.
"And if some guys find some form and score 80, 90 or 100, that's probably the positives we can take out of it."
The visitors resumed after seizing control on day two when Australia were dismissed for a woeful 104 in reply to India's 150.
Jaiswal, who began on 90, reached three figures with an extraordinary six off Hazlewood, hitting the boundary marker on the full, triumphantly raising his arms in the air.
It was his fourth ton and first in Australia.
Mini collapse
Since scoring 171 on debut against the West Indies in 2023 Jaiswal has cemented his place at the top of the Indian order.
Two double centuries at home against England this year catapulted him to stardom, but concerns lingered over how he would perform in Australian conditions.
After being out for a duck in the first innings his response against one of the world's best attacks put any doubts to bed.
With the sun beating down, Australia finally earned some reward when Rahul feathered Mitchell Starc to wicketkeeper Alex Carey.
Playing in place of Rohit Sharma, who stayed home for the birth of his second child, the 32-year-old did a stellar job supporting his young partner.
Skipper Rohit is set to return for the second Test in Adelaide, posing a selection headache for the visitors.
Australia got another breakthrough on the first ball after lunch when Padikkal edged Hazlewood to Smith at slip.
Jaiswal's dismissal sparked a mini collapse with Pant stumped by Carey and Jurel out lbw to Pat Cummins.
But Kohli and Sundar steadied the ship, putting on 89 for the sixth wicket, before Nathan Lyon bowled Sundar.
That introduced Nitish Kumar Reddy, who with Kohli went on an onslaught against a tired attack.