The visitors took a 46-run lead into the second innings after dismissing the hosts for a meagre 104 at lunch in reply to their equally miserly 150 on a lively pitch.
By tea, Jaiswal was not out 42 and Rahul on 34 as they worked hard to put the match out of Australia's reach and draw first blood in the five-match series.
Jaiswal, in his 15th Test, failed to score in his first knock and began tentatively before a boundary off Mitchell Starc settled him.
Batting alongside the more experienced Rahul, they ran well between the wickets and cracked anything loose to the boundary, playing their shots as their confidence grew.
It included a beautiful straight drive by Rahul off Pat Cummins and an equally impressive sweep by Jaiswal from a fullish Starc delivery.
The ball was not moving nearly as much as the opening day and Australia were in dire need of some inspiration, but it failed to come as the partnership flourished.
After an astonishing 17 wickets fell on a chaotic opening day, Australia resumed on 67-7 and put on 37 thanks to a dogged last wicket holdout.
They reached three figures courtesy of Starc and Josh Hazlewood, with their 25-run stand the longest of the Australian innings.
Jasprit Bumrah was India's chief destroyer with 5-30, his 11th five-wicket haul in Tests. Harshit Rana chipped in with 3-48.
Home hopes rested on Alex Carey as he began on 19, but after nudging two from Rana he came up against an irrepressible Bumrah.
Coming round the wicket, the Indian captain made Carey play and he nicked to wicketkeeper Rishabh Pant.
Nathan Lyon (5) did well to hang around for 16 fiery balls before gloving a short one from Rana to Rahul in the slips.
That brought Hazlewood to the crease and it seemed only a matter of time before India were batting again.
But he and Starc gamely stuck around, bringing up the 100 to huge cheers from the crowd.
It was an intelligent innings by Starc, who doggedly faced 112 balls and protected Hazlewood to add crucial extra runs before holing out Rana to Pant on 26.
India have won their last two Border-Gavaskar trophy series in Australia, but came into the Perth clash after a humiliating 3-0 home loss to New Zealand.
