Bangladesh left-arm spinner Taijul Islam claimed 4-47 to complete his first career 10-for on Monday while Zimbabwe continued to dominate on the third day of the first cricket Test.
At tea, the visitors reached 6-165 in their second innings, taking their lead to 304.
Bangladesh's highest successful fourth innings chase is 6-217 against the West Indies at St. George's in 2009.
Regis Chakabva was batting on 19 with Wellington Masakadza on 17, frustrating Bangladesh, which got wrapped up Zimbabwe cheaply following Taijul's three wickets in his five balls after lunch.
They played 111 balls between them to add an undefeated 35 runs for the sixth wicket stand so far.
Offspinner Mehidy Hasan took 2-48.
Mehidy broke through dismissing opener Brain Chari for four to give the hosts a hope to fight back in the morning session. But Brendan Taylor counterattacked in a bid to keep Bangladesh's spinners at bay.
Islam ended Taylor's attacking knock by inducing him to play a big shot, resulting in a catch at long-on.
Taylor scored 24 off 25, hitting four boundaries.
Mehidy, however, got the prized wicket of Hamilton Masakadza for 48, ending his 104 balls patient knock before Taijul ran through the middle order.
Bangladesh were bowled out for 143 in the first innings, their seventh straight Test innings below 200, in reply of Zimbabwe's 282.
