Miraz remained unbeaten on 109 off 141 balls, hitting 11 fours and four sixes, as Bangladesh recovered from a shaky start to post 263 in 75.5 overs.
The visitors then reduced Cricket Australia XI to 51 for 2 in 12 overs, leaving the hosts 212 runs behind at the close.
Bangladesh's innings got off to a poor start as Tanzid Hasan was dismissed for a duck before the top order struggled to build substantial partnerships.
Shadman Islam made 31, Mominul Haque scored 16, captain Najmul Hossain Shanto contributed 37, while Mushfiqur Rahim was dismissed for a duck. Young wicketkeeper-batter Amite Hasan added 15 before Miraz steadied the innings with a composed century.
Soumya Sarkar chipped in with 16, Taijul Islam made 10 and Khaled Ahmed added a useful 15 late in the innings.
Off-spinner Corey Rocchiccioli starred for Cricket Australia XI with outstanding figures of 6 for 83, while Hanno Jacobs, Xavier Crone, Campbell Thompson and Jerrssis Wadia claimed one wicket each.
In reply, Bangladesh's fast bowlers struck twice in the opening three overs. Hasan Mahmud removed Sam Konstas for one, while Taskin Ahmed dismissed Campbell Kellaway for a duck to leave the hosts reeling at 1 for 2.
Kurtis Patterson then led the recovery with an unbeaten 26, sharing an unbroken 50-run stand with Josh Philippe, who was not out on 22, as Cricket Australia XI reached 51 for 2 at stumps.
Hasan finished the day with figures of 1 for 4 from four overs, while Taskin claimed 1 for 24.
