Bumrah picked Usman Khawaja very early in the day after the Aussies resumed play at 12/3.
After Khawaja departed, veteran Steve Smith and Travis Head resisted the Indian bowlers, stitching a 62-run partnership.
However, Siraj picked the second wicket of the day, sending Smith back to the hut after the right-handed batter scored a patient 17 from 60 balls.
At lunch on day 4, Australia crawled to 104/5, still needing 430 runs to overwhelm India in the match, which was majorly dominated by the guests.
India on Sunday ended their second innings at a mammoth score of 487/6 declared, riding on young Yashasvi Jaiswal's 161 and veteran Virat Kohli's unbeaten 100 which saw the batter's strong comeback after a dull period.