The home side trail by 224 runs against Bangladesh, who posted 468 in their first innings with Mahmudullah Riyad (150 not out) and Liton Das (95) posting Test career bests.
Entering the third and final session of the day, Zimbabwe needed 25 more runs to prevent Bangladesh captain Mominul Haque having the option of enforcing a follow-on.
Zimbabwe opener Takudzwanashe Kaitano was at the crease having crafted a patient 82 off 282 balls, including nine fours, in warm mid-winter conditions.
Kaitano was partnered by Regis Chakabva, who exercised extreme caution as he contributed 10 runs off 32 deliveries.
Shakib al Hasan inflicted most second-session damage for Bangladesh, taking the wickets of debutant Dion Myers (27) and Timycen Marumo (0).
A Myers sweep was caught by Mehidy Hasan Miraz at fine leg and Marumo was trapped leg before having failed to score off 17 balls.
Taskin Ahmed claimed the other second-session wicket, ending a run-less five-ball stay at the crease by Roy Kaia.