Tigers fought well when Shakib Al Hasan and Tamim Iqbal were at the crease but Australia spinners have been able to break the partnership as the duo returned to the pavilion.
The hosts were 3-96 at lunch at the Sher-E-Bangla stadium, with Shakib Al Hasan unbeaten on 48 and Tamim Iqbal on 33.
The 86-run stand between danger men Shakib and Tamim was much-needed after Cummins left Bangladesh reeling at 3-10 off four overs.
Tamim Iqbal departed after scoring his 23rd test half century. Shakib edges a flighted turner from Nathan Lyon and is caught by captain Steve Smith at slip. Shakib tentatively pushes at the delivery and is out for 84.
Cummins took three wickets in eight balls, the tearaway accounting for three of the hosts' top four in about 20 minutes.
The right-armer sent an early warning to the Bangladesh dressing room as a he got one to rise sharply on opener Soumya Sarkar, whose edge just eluded the grasp of Usman Khawaja at third slip.
Two balls later, Cummins had his man as Sarkar edged another lifter and Peter Handscomb pouched a simple chance in the gully.
First-drop Imrul Kayes went the following over, the left-hander playing away from his body and inside-edging behind to wicketkeeper Matthew Wade.