Bangladesh will require seven more wickets on the final day to seal their maiden Test win, after Zimbabwe closed 46-3, 335 runs short of their target.
The Tigers captured two wickets in the opening four overs with only two runs scored, and removed opener Stuart Matsikenyeri six overs before stumps.
Habibul Bashar made 55 from 80 balls as Bangladesh declared on 204-9.
Zimbabwe's last two first innings wickets had earlier added just four runs as they posted a total of 312.
Wicket-keeper Khaled Mashud caught both Mluleki Nkala and Hondo as Bangladesh took a 176-run first innings lead.
" I am 100% confident of winning but we have to take seven wickets in 90 overs. "
But the home side soon suffered a familiar batting collapse, losing six wickets for 90 in 25.4 overs as they pressed for quick runs.
After having Aftab Ahmed caught at mid on for 11, Elton Chigumbura captured the prize wicket of Habibul, who hit six fours.
The rangy medium-pacer finished with 5-54, his first five-wicket haul in his third Test but Bangladesh were suitably content with their lead to do without the batting talents of last man Enamul Haque Jr.
Tapash Baisya took wickets in both his opening overs as Zimbabwe made a woeful start in their pursuit of an unlikely 381.
The unfortunate Barney Rogers, run out at the non-striker's end for five in the first innings, fell at second slip for a duck, while Vusimuzi Sibanda was trapped lbw sixth ball, also without scoring.
Haque then ended a stand of 40 in 13.2 overs between Matsikenyeri and Hamilton Masakadza when the opener played across the line and the ball hit both middle and off stumps.
Masakadza, who remained undefeated after hitting five fours in his 26, accepted the offer of bad light with one scheduled over remaining.
Bangladesh: Habibul Bashar (captain), Khaled Mashud, Nafis Iqbal, Javed Omar, Mohammed Ashraful, Rajin Saleh, Aftab Ahmed, Mohammed Rafique, Enamul Haque, Tapash Baishya and Mashrafe bin Mortaza.
Zimbabwe: Tatenda Taibu (captain), Elton Chigumbura, Stuart Matsikenyeri, Barney Rogers, Vusimuzi Sibanda, Hamilton Masakadza, Brendan Taylor, Graeme Cremer, Christopher Mpofu, Douglas Hondo, Mluleki Nkala.
Umpires: T Wijewardena (Sri), Asad Rauf (Pkn)