At tea the visitors were 321-5, after rain washed out the entire first session at the Zahur Ahmed Chowdhury stadium in Chittagong.
Australia now lead by 16 runs with five wickets in hand in their first innings. Bangladesh, who elected to bat first, made 305.
Resuming the day at 225-2, Warner and Peter Handscomb shared 152 runs for the third wicket to frustrate the Bangladesh bowlers.
Warner showed remarkable patience during his uncharacteristically sedate 123-run knock that came off 234 balls, containing seven fours.
He was dismissed by left-arm paceman Mustafizur Rahman, who was celebrating his 22nd birthday today.
The century saw Warner join a select group of 12 Australian cricketers who have scored 20 or more Test centuries, with his skipper Steven Smith the most recent to previously achieve the feat.
Warner, who was unbeaten on 88 overnight, received good support from Handscomb, who made 82 off 144 balls before being run out by a sharp throw from Shakib Al Hasan at midwicket.
Bangladesh were then frustrated as Mehedi Hasan dropped Glenn Maxwell on 10 at gully off Mustafizur.
Mehedi then put down Hilton Cartwright off his own bowling on six, before making amends by having the same batsman caught at slip by Soumya Sarkar for 18 on the stroke of tea.
Maxwell was unbeaten on 25 at the break.
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