At tea on day three the visitors were 359-5, a commanding 405-run lead as they eye victory and a psychological advantage in the five-Test series.
Virat Kohli was not-out 40 and Washington Sundar on 14.
Jaiswal, who grew up homeless in the slums of Mumbai selling snacks on the streets to survive, was the hero, hitting 15 fours and three sixes in a mature 297-ball stay.
The 22-year-old, in his 15th Test, seemed destined for a second career double century but he cut Mitchell Marsh to Steve Smith at point and departed to a standing ovation.
Australia also removed Devdutt Padikkal (25), Rishabh Pant (1), and Dhruv Jurel (1), 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.
The visitors resumed with a 218-run advantage after seizing control on day two after Australia were dismissed for a woeful 104 in reply to India's 150.
Padding up on 172-0, Jaiswal took a risky single off Josh Hazlewood's opening ball to signal their attacking intent before two boundaries from Mitchell Starc's first over.
Left-hander Jaiswal, who began on 90, reached three figures with a 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.
Since scoring 171 on debut against the West Indies in 2023 Jaiswal has cemented his place at the top of the order.
An eye-opening 214 not out 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 for their toil when Rahul feathered 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 headache for selectors.
Australia got another breakthrough on the first ball after lunch when Paddikal edged Hazlewood to Smith at slip.
Jaiswal's dismissal sparked a mini collapse with three wickets for eight runs before Kohli and Sundar steadied the ship.
Pant charged down the pitch to Nathan Lyon and was stumped by Carey while Jurel fell lbw to Pat Cummins.
