After dismissing Australia for 104 in 51.2 overs in the morning session to take a 46-run first-innings lead, Jaiswal (90 not out, 193 balls) and Rahul (62 not out, 153 balls) scored freely as the visitors put themselves in a strong position by swelling their lead to 218 runs.
Earlier, pacer Jasprit Bumrah completed his 11th five-wicket haul in Tests as Australia were bowled out soon after reaching the century mark at lunch.
For India, Bumrah (5/30) was the pick of the bowlers while Mohammed Siraj (2/20) and debutant Harshit Rana (3/48) also chipped in.
It was an abject surrender from Australia, who could add just 37 runs to their overnight score of 67 for 7, thanks chiefly to tail-ender Mitchell Starc (26), who emerged the highest scorer for his side.
Brief Scores: India: 150 & 172 for no loss in 57 overs (Yashasvi Jaiswal 90, KL Rahul 62) vs Australia: 104 all out in 51.2 overs (Alex Carey 21, Mitchell Starc 26; Jasprit Bumrah 5/30, Harshit Rana 3/48, Mohammed Siraj 2/20).
