Spinner Mehidy Hasan Miraz claimed five wickets as Bangladesh managed to restrict Zimbabwe to 273 on Monday for a modest 82-run lead on the second day of the first Test.
Zimbabwe finished the first day in Sylhet in a commanding position after bowling out the hosts for 191 and then racing to 67-0.
However, opener Ben Curran fell on 18 to Nahid Rana at the start of the second day and his partner Brian Bennett, who hit an aggressive 64-ball 57, was removed soon after with the score on 88-2.
Sean Williams (59) was the only other Zimbabwe batter to reach fifty. Wessly Madhevere (24), Nyasha Mayavo (35) and Richard Ngarava (28) all failed to build bigger scores.
Off-spinner Mehidy took the key wickets of Williams, Mayavo and Ngarava to finish with 5-52 on a wicket that was expected to help the pace bowlers.
Nahid took 3-74 with aggressive bowling in support of Mehidy.
Bangladesh skipper Najmul Hossain Shanto won the toss on the first day and opted to bat first but his batsmen were unable to capitalise.
Shanto must now hope that they can make the most of the friendly home conditions to set a formidable target for the visitors.
Bangladesh have won eight of the 18 Tests against Zimbabwe, their highest total against any Test side, with four of them in their past five encounters.
The second and final Test of the series will be played in Chattogram from 28 April.