Australia reached tea on 51 for 2 in their second innings, with Hasan Mahmud accounting for both openers before Steve Smith and Marnus Labuschagne steadied the innings, leaving the hosts still 177 runs behind after Bangladesh's first-innings total of 426.
Bangladesh's innings had ended just after lunch, bowled out for 426 with Josh Hazlewood removing Ebadot Hossain to complete figures of 6 for 89 and bring up his 300th Test wicket, becoming only the ninth Australian bowler to reach the milestone.
Hasan struck twice with the new ball to give Bangladesh an ideal start in the field. Jake Weatherald fell without scoring in the second over, bowled outright, before Travis Head edged an attempted cut onto his stumps for 17, giving Hasan his eighth wicket of the match across both innings.
With that pair of dismissals, Hasan became only the second Bangladeshi bowler to remove both opposition openers in each innings of the same Test, following Robiul Islam's effort against Zimbabwe back in 2013.
Smith and Labuschagne have since begun to settle, adding an unbroken stand that took Australia through to the break with minimal alarm, with Smith unbeaten on 8 and Labuschagne on 21.
Smith survived an early scare when he reviewed a caught-behind decision before the umpire had even raised his finger, correctly sensing the ball had come off the ground rather than his bat, while Labuschagne top-edged a pull that dropped just short of the man at midwicket, before both settled to work the ball around with growing confidence.
The Darwin surface has offered far more for the batters on this occasion than the grassy, seam-friendly pitch Australia batted on first.
Bangladesh's total of 426 stands as their second-highest in Tests against Australia, just short of the 427 they made in the first innings of the 2006 Fatullah Test, built on Tanzid Hasan Tamim's century, Najmul Hossain Shanto's 84, and half-century from Mehidy Hasan Miraz among other contributions.
Bangladesh will know the job in this Test is far from finished, even with a substantial lead already banked and further wickets needed to keep control of the contest heading into the final two days.