After the first innings of each team, Zimbabwe maintained an 82-run first innings lead.
Bangladesh lost their first wicket early in the second innings when Shadman Islam fell prey to Blessing Muzarabani. Bangladesh ended the second day at 57 for one.
Mahmudul Hasan Joy, who was unbeaten overnight, could only manage five runs on day three before falling as the second wicket of Muzarabani.
After the fall of both openers, Mominul and Shanto paired up to guide Bangladesh.
At the end of 25 overs, Bangladesh were at 105 for two.
Earlier, Zimbabwe made 273 in their first innings. Mehidy Hasan Miraz took a five-for, while Nahid Rana took three.
Bangladesh posted a 191 in their first innings after winning the toss. Mominul scored a fifty for the hosts, but the other batter failed to capitalise, reports UNB.
The second and final Test of this series will be played in Chattogram on April 28.