Opener Sikandar Raza remained unbeaten on 54 and Hamilton Masakadza was on 51 as the tourists reached 113-1 by stumps, still trailing by 390 runs with nine wickets in hand.
The pair added 104 for the unbroken second wicket after opener Brian Chari was caught behind off Rubel Hossain without scoring.
Rubel had earlier shone with the bat, making an entertaining 45 off 44 balls to help Bangladesh post their highest total against Zimbabwe, surpassing the 488 in Chittagong in 2005.
Openers Tamim Iqbal (109) and Imrul Kayes (130) had laid the platform for a big total on Wednesday and Shakib Al Hasan led Bangladesh's batting charge on the second day with 71.
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Zimbabwe's bowlers prevented a bigger total by claiming eight wickets in the first two sessions after the hosts resumed the day at a commanding 303-2.
Mominul Haque added just two runs to his overnight score of 46 when he was caught the slips by Brendan Taylor off Tinashe Panyangara.
The hosts are hoping to sweep the series 3-0 after winning the first two Tests in Dhaka and Khulna.
The Test series between the two lowest-ranked teams will be followed by five one-day internationals.