Zimbabwe opener Takudzwanashe Kaitano dug in for a gritty 87 as the home side saved the follow-on at the Harare Sports Club on Friday but Bangladesh ended the third day of the one-off test well in control.
After bowling out Zimbabwe for 276 to claim a first-innings lead of 192, the Bangladesh openers batted through to the close to reach 45 without loss and extend that lead to 237.
Shadman Islam was 22 not out and Saif Hassan unbeaten on 20.
"When we came to fielding, the captain asked us (to) give less runs, Shakib also told (me)," said offspinner Mehidy Hasan Miraz, who took five test wickets for the eighth time.
"They (Zimbabwe) already made a big partnership, (but) we bowled well after lunch. Pacer, spinners all put in an effort to help us get success.
"The wicket was flat today. It was a tough job for our spinners. But we tried to bowl in one area, stop their run flow and make them play wrong shots to our good deliveries. The strategy for spinners worked.
"I got five wickets after many days. I was getting wickets, but only two or three in an innings… it gives confidence to any bowler.
"I got five wickets (abroad) in the West Indies. Now it feels good to get five in Zimbabwe."
The Zimbabwe innings revolved around two innings of contrasting styles, Kaitano's brave and patient 87 and a swashbuckling 81 from stand-in captain Brendan Taylor.
ZIMBABWE TAIL COLLAPSES
Test debutant Kaitano faced 311 balls before being caught down the legside by wicketkeeper Liton Das, trying to flick Mehidy away through midwicket.
After the departure of Kaitano with the hosts 261-6, the Zimbabwe tail collapsed, adding only 15 runs.
Day three had begun promisingly for the home team with a boundary-littered innings by Taylor lifting his side to 209-2 by lunch.
Taylor and Kaitano added 115 for the second wicket and appeared to be in serene control, Kaitano's stout defence complemented by the skipper's more robust approach.
Taylor faced 92 balls and struck 12 fours and one six before miscuing a slog off Mehidy and spooning the ball to substitute Yasir Ali at square-leg.
It was a crucial moment as it changed the mood and momentum of the match, culminating in a collapse that saw eight wickets go down for just 51 runs as Zimbabwe slumped from 225-2 to 276 all out.
Mehidy and his fellow bowlers had toiled throughout the first session, conceding 95 runs and capturing just one wicket.
But there was a dramatic change after lunch as Zimbabwe added only 35 runs off 29 overs and lost three wickets during the second session as they crept to 244-5 by tea.
Shakib al Hasan did most of the damage, removing debutant Dion Myers, who made 27, and Timycen Marumo for a duck.
A Myers sweep was caught by Mehidy at fine leg and Marumo was trapped leg before having failed to score off 17 balls.
Taskin Ahmed claimed the other second-session wicket, finding the edge of Roy Kaia's bat as Das pouched another catch to send him back, also for a duck.
After Kaitano's dismissal, only dogged Regis Chakabva (31 not out) offered any sustained resistance.
ZIMBABWE: Takudzwwanashe Kaitano, Milton Shumba, Brendan Taylor (capt), Dion Myers, Timycen Maruma, Roy Kaia, Regis Chakabva (wk), Donald Tiripano, Victor Nyauchi, Blessing Muzarabani, Richard Ngarava
BANGLADESH: Shadman Islam, Saif Hassan, Mominul Haque (capt), Najmul Shanto, Mushfiqur Rahim, Shakib al Hasan, Mahmudullah Riyad, Liton Das (wk), Mehidy Hasan, Taskin Ahmed, Ebadot Hossain