Hasan became the first Bangladeshi player to score a century in Australia, helping the Tigers to a 153-run lead at the close of play, with four wickets still in hand on a flattening pitch.
Bangladesh resumed play on a sunny day two at 96/1 and the hosts had to be patient for the first breakthrough. The watchful Mominul Haque and Hasan combined for 102 before Josh Hazlewood finally broke the partnership. The big quick drew Haque into a drive with a full and wide delivery, only to get a thick outside edge to Carey for 49.
That brought captain Najmul Hossien Shanto to the crease, and he picked up where Haque left off. The patient pair frustrated the Australians, poking and prodding their way to lunch.
Shanto became more expansive after the break, reaching his fifty off the second ball of spinner Nathan Lyon's over before the drinks break. Hasan - whose cover drive was picture perfect all day - had his moment in the sun shortly after the break, driving Lyon down the ground to bring up his ton and cue celebrations in the Bangladeshi camp.
But as Hasan looked to press on, he miscued a heave over Lyon and was caught in the deep, signing off for a brilliant 101 in only his second Test and against the world top Test nation to boot.
Australia looked forlorn as Bangladesh went to tea at 286/3. But the new ball finally brought joy as three wickets fell for just 11 runs. Hazlewood was first to strike, drawing an edge from Shanto (84) to second slip. It was a carbon copy for Mitchell Starc to get rid of Litton Das for a duck, and again for Pat Cummins as the dangerous Mushfiqur Rahim departed for 36. Veteran Steve Smith bagged all three, which took him to an incredible 218 catches in Tests, tied at the top with Joe Root.
With their tails up, Australia's quicks continued to charge in but Mehidy Hasan Miraz and Hasan Mahmud weathered the late storm. Their stoic batting even forced part-timers Travis Head and Marnus Labuschagne to turn their arms over, to no avail.
To rub salt in the wound, a wicketless Cameron Green dropped a low one at first slip on the final ball of the day. That meant Miraz and Mahmud had survived 147 balls for their unbeaten partnership of 43, finishing on 351/6.
Joe Root has admitted he is "really hacked off" by Brydon Carse's latest off-field controversy, insisting England must rebuild trust after the fast bowler was withdrawn from the squad for the second Test against Pakistan.
England fast bowler Brydon Carse has been withdrawn from the squad for the second Test against Pakistan after an incident at a nightclub sparked an investigation by the Cricket Regulator.
Dhruv Jurel struck the second Test century of his career as India strengthened their grip on the second Test against Sri Lanka.
Pat Cummins has welcomed Australia's long-awaited return to South Africa, describing the Proteas as a "really strong side" as the visitors begin preparations for a three-Test series in October.
