Taijul Islam pulled a short delivery towards backward square leg boundary to bring up the winning runs at the Sher-e-Bangla National Stadium in Mirpur on Monday.
Bangladesh now lead the three-match series 1-0. The second Test is scheduled for Nov 3 at the Sheikh Abu Naser Stadium in Khulna.
On a wet surface, the hosts started their chase of 101 to win in the worst possible fashion losing three wickets without being able to score a single run.
The disaster sucked away the buoyant mood after Taijul decimated the Zimbabwe batting order with 8 for 39, the best figures by a Bangladeshi bowler, to skittle Zimbabwe out for 114 in their second innings.
Tamim Iqbal edged Elton Chigumbura to Brendan Taylor at second slip to trigger the collapse.
The Mirpur crowd was hushed into silence when Shamsur Rahman and Mominul Haque went back to successive deliveries.
Mahmudullah and Shakib Al Hasan came together to add 46 runs for the fourth wicket to bring some relief to the Bangladesh camp.
Shakib (15), dropped on 2, scored off short balls but eventually fell to Tendai Chetara.
The hosts again slipped back into trouble when Mahmudullah (28) and Shuvagata Hom Chowdhury went back in a Chigumbura double-wicket maiden over.
With Bangladesh stuttering at 62 for 6, captain Mushfiqur Rahim and Shahadat Hossain put up resistance.
Shahdat was being watchful but right after bludgeoning Tinashe Panyangara for a six he was sent packing to put the Tigers under more pressure.
They still required 19 runs to win.
But Mushfiqur saw off the rest of the chase with Taijul, in at nine, with ease. The skipper remained unbeaten on 23 and Taijul on 15.
Chigumbura was the pick of the bowlers for Zimbabwe with 4 for 21.
But it was Taijul who rocked the Africans with five wickets in the first session of the day.
The left-arm orthodox spinner became the first Bangladesh bowler to perform the feat.
The tourists, who were 5 for 0 at stumps on Day Two, lost wickets of opener Vusi Sibanda (14) and Hamilton Masakadza (5) first up. The former became Taijul’s first scalp.