The match was rain interrupted by more than 1 1/2 hours due to a wet outfield and reduced to 18 overs per team.
Zimbabwe which was put into bat by the Bangladeshi skipper, posted an imposing 144-5 and looked certain to pull off a victory with Bangladesh 60-6 in the 10th over. But 19-year old Afif Hossain, playing only his second match, counter-attacked in with a quick-fire half ton as Bangladesh reached 148-7 in 17.4 overs.
Afif and Mosaddek Hossain combined for 82 runs off 47 balls in their seventh-wicket partnership to take Bangladesh close to victory before Afif was dismissed by Neville Madziva (2-25) in the last over of the game.
Afif, who brought up his 50 off 24 balls, struck eight fours and one six. Mosaddek was not out on 30 off 24, including two sixes.
Zimbabwe pacer Tendai Chatara took 2-32, including the wicket of opener Liton Das for 19. Fast bowler Kyle Jarvis (2-31) dealt a double blow, removing Soumya Sarkar (4) and Mushfiqur Rahim (0) before Chatara took the prized wicket of Shakib Al Hasan for 1.
And when Ryan Burl had Mahmudullah lbw for 14, Bangladesh was in real danger. The situation worsened with the dismissal of Sabbir Rahman for 15, leaving the hosts 60-6.
Earlier, Burl hit a 32-ball 57 not out for Zimbabwe with five fours and four sixes, including smashing 30 runs off Shakib's last over - 6, 4, 4, 6, 4, 6 - to help the side post a challenging total. It was the highest number of runs conceded by Shakib in his T20 career and he finished with 4-0-49-0, also his worst bowling figures.