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Four wickets fell in an eventful second session but Kohli and Washington Sundar (14) remained firm during the breaks as India looked in commanding position.
Jaiswal, who completed his fourth Test hundred in the morning session, was caught by Steven Smith off Mitchell Marsh. He had 15 boundaries and three maximums in his 297-ball innings.
Devdutt Padikkal (25), Rishabh Pant (1) and Dhruv Jurel (1) were the other Indian batters to be dismissed.
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In their second innings, Jaiswal and KL Rahul (77) stitched a record-breaking 201-run stand for the opening wicket, the highest by an Indian opening pair on Australian soil, to put India in a strong position.
Brief Score: India: 150 & 359 for 5 in 110 overs (Yashasvi Jaiswal 161, KL Rahul 77, Virat Kohli 40 batting; Josh Hazlewood 1/28, Mitchell Marsh 1/35, Nathan Lyon 1/58, Mitchell Starc 1/102) vs Australia: 104.