Shakib Al Hasan followed up his half-century on the 50th Test match on Sunday with a five-wicket haul on the second day Monday.
Australia could have been in deeper trouble but for the rearguard defiance of Pat Cummins and Ashton Agar, who raised 49 runs for the ninth wicket to lend a touch of respectability to their total.
Bangladesh went to stumps on 45 for one to hold an overall lead of 88 after bowling Australia out for 217 shortly after tea.
Starting off the day on 18/3, Australia only added 15 runs to their overnight score before captain Steve Smith was clean bowled by Mehidy Hasan Miraz for eight runs.
Maxwell reached 23 before being stumped by Bangladesh skipper Mushfiqur Rahim, after charging down the wicket and completely missing a Shakib delivery.
Matt Renshaw and Peter Handscomb were also subjected to a trial by spin with Mehidy (3-52) and left-arm spinner Shakib Al Hasan (3-54) hunting in tandem on a surface that perfectly suited their craft.
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He said he expected to receive a hefty suspension, but was angry that the Yankees were not also punished. Both Tigers manager Brad Ausmus and Yankees manager Joe Girardi were ejected in the contest as well.
Tamim Iqbal (30) and nightwatchman Taijul Islam, who is yet to get off the mark, will resume on the third day trying to build the home team's lead. Shakib Al Hasan has joined Rangana Herath, Dale Steyn and Muttiah Muralitharan in this elite club.
Renshaw survived nearly two sessions, facing 94 balls for his 45 runs while Australia lost five wickets, until he edged to Soumya Sarkar at slip.
Yet Lyon's landmark day was soured as he went on to make a duck after being sent in with Australia two down.
Australia got some much-needed luck in the form of a howler of a dropped catch from Bangladesh's Shafiul Islam after Cummins skied one from Shakib on 11.
"But on 25 from 89 balls, Cummins tried to drive Shakib through the off side and was bowled; Agar finished unbeaten on 41 off 97 when Josh Hazlewood was taken at short leg off Shakib for 5", wrote Brydon Coverdale in his report.
Shakib returned figures of 5-68 to help his side dismiss Australia for 217. He seemed to confer at length with non-striker Glenn Maxwell before opting not to review the decision despite Australia having two challenges up their sleeve. But if Australia thought Bangladesh would be pushovers, they reckoned without Shakib and Tamim, who put together the second-highest fourth-wicket partnership in Bangladesh's Test history to lead the recovery. "No matter in what manner the wicket behaves, unless you can bowl in the right areas it is hard to earn success".