The hosts, replying to Zimbabwe's first-innings total of 240, were dismissed for 254 just before stumps on the second day at the Sher-e-Bangla stadium.
Zimbabwe were five for no loss in the second innings at close of play, needing a competitive total to challenge Bangladesh on a wearing wicket.
Skipper Mushfiqur top-scored for Bangladesh with 64, Mahmudullah scored 63 and Mominul chipped in with 53 in overcast conditions that forced the floodlights to be switched on.
Bangladesh were restricted by three run outs and a maiden five-wicket haul from seamer Tinashe Panyangara, who finished with 5-59.
Resuming at their overnight score of 27-1, Bangladesh lost Shamsur Rahman in the third over of the day when he spooned a catch to Elton Chigumbura at mid-off off Panyangara.
Mominul, who shared a third-wicket stand of 63 with Mahmudullah, was run out before the lunch break by a direct throw from Sikandar Raza in the cover region.
Shakib Al Hasan too was run out soon after lunch for five, while Mahmudullah was adjudged leg-before off part-time spinner Raza when the appeal was referred to the third umpire. Mahmudullah was initially adjudged not out, but replays showed the ball hitting his leg-stump.
Mushfiqur batted solidly till he miscued a pull off Panyangara’s bowling to Craig Ervine at square leg.
Shahadat Hossain was the third batsman to be run out and Bangladesh managed to take the lead only after number eight Taijul Islam scored 19.
Panyangara bowled both Taijul and number 11 Al-Amin Hossain to complete his five-wicket haul.
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