Bangladesh's first innings finally ended on 426 all out shortly after lunch on the third day in Darwin.
Josh Hazlewood removed last man Ebadot Hossain, caught behind for 7, to complete figures of 6 for 89 and hand Bangladesh a commanding first-innings lead of 228 runs over Australia. Taskin Ahmed remained unbeaten on 14.
The total stands as Bangladesh's second-highest against Australia in Tests, just short of the 427 they made in the first innings of the Fatullah Test back in 2006.
Mehidy Hasan Miraz anchored the innings from the middle order, resuming on an overnight 32 and grinding his way to 65 off 152 balls with five fours before falling caught behind off Hazlewood, who reached his 300th Test wicket in the process.
Hasan Mahmud, promoted by circumstance into a lengthy rearguard role, faced 92 balls for 14 before being trapped lbw by Hazlewood, ending a 46-run seventh-wicket stand with Miraz that had eaten up 170 deliveries and frustrated an increasingly weary Australian attack.
Taijul Islam then chipped in with a breezy 17 off 23 balls, including a six struck over deep backward square leg, before falling at gully off Hazlewood.
Taskin Ahmed joined Miraz for a defiant ninth-wicket stand of 45 that pushed Bangladesh past the 400 mark and deep into record territory.
Australia's frustration showed in the field, with Steve Smith and Travis Head both shelling regulation chances late on, Smith missing a simple catch that would have made him the sole holder of the record for most catches by a fielder in Tests.
Miraz eventually fell to end the last recognised resistance, before Hazlewood mopped up the tail.
Bangladesh batted 138 overs in all, comfortably their longest innings against Australia and one of only 17 occasions they have batted beyond 120 overs in an innings away from home.
The platform for the total was laid on the second day, when Tanzid Hasan Tamim became the first Bangladeshi batter to score a Test century on Australian soil.
Resuming on 96 for 1, he added a maiden Test fifty before completing his century off 188 balls in the 64th over, an innings of 101 built on eight fours and a six before he fell attacking Nathan Lyon.
Tanzid's dismissal ended a 92-run stand with captain Najmul Hossain Shanto, who went on to make 84 off 127 balls with the bat flowing freely, while Mominul Haque had earlier contributed a well-made 49.
Mushfiqur Rahim's 36 and a brief middle-order wobble against the second new ball were the only real hiccups in an otherwise dominant day.
It was Hasan Mahmud's bowling on the opening day, however, that set the tone for the entire Test.
Bangladesh bowled Australia out for just 198 after being sent in to field first, with Hasan claiming career-best figures of 6 for 55 and Ebadot Hossain and Taskin Ahmed sharing the remaining four wickets between them.
Steve Smith's 71 was the lone significant resistance for the hosts, an innings given a second life when Tanzid dropped him early off Ebadot, but Bangladesh's pace trio otherwise ran through the order with control.
It marked the first time Bangladesh had bowled Australia out in a Test on Australian soil, in the country's first Test there in 23 years.
Australia's history in similar positions offers little comfort. The last time they won a Test after trailing by more than 100 runs on first innings came 16 years ago, against Pakistan, and they have overturned first-innings deficits of more than 200 runs only three times in 82 previous instances, leaving Bangladesh firmly in control heading into the second innings.