Bangladesh went to lunch on the second day in Mackay at 39 for 3, trailing by 107 runs, with Mitchell Starc again causing early damage as captain Najmul Hossain Shanto stood unbeaten on 23 and Mushfiqur Rahim yet to get going on 4.
Shanto had looked the most assured batter at the crease, striking a boundary off just his second ball and continuing to find gaps with confidence throughout the session.
He survived one scare against Nathan Lyon, when Alex Carey's appeal for a catch behind was sent upstairs by Pat Cummins, only for Ultra Edge to show no contact and cost Australia their first review.
Bangladesh's innings had started poorly well before that. Starc struck twice in successive deliveries early on, with Shadman Islam edging behind to Marnus Labuschagne at point for just one, the third time this series he has fallen in near identical fashion, before Mominul Haque was bowled first ball for a golden duck.
Just as Shanto and Tanzid Hasan looked to be steadying the innings, Tanzid fell into a short ball trap Australia had clearly set.
Hooking a Cummins bouncer straight to long leg, he was caught by Starc for 9 from 34 balls, having faced little trouble until that point.
The platform for Australia's advantage had been laid earlier in the morning, when the tail added a stubborn 45 runs before finally folding.
Nathan Lyon finished unbeaten on 32 from 62 balls, striking four fours and a six, and shared a defiant 24 run last wicket stand with Josh Hazlewood that included 17 runs off a single Taskin Ahmed over.
Taijul Islam eventually ended the innings, having Hazlewood caught by Shadman at midwicket as he went for another big shot, with Australia bowled out for 210 in 52.3 overs to take a 146 run first innings lead.
That lead was built almost entirely on the back of Shoriful Islam's collapse of the Australian middle order a day earlier, anchored by a searing yorker that trapped Cameron Green in front.
His figures of 7 for 48 stand as the best innings return by a Bangladesh bowler overseas in Test cricket, surpassing Taijul's own 7 for 138 in Harare, and the best by a visiting bowler in Australia since 1993.
Taskin, Taijul and Mehidy Hasan Miraz shared the remaining three wickets between them.