Chasing a modest target of 144, Bangladesh reached 144/6 in 19.4 overs, with opener Tanzid Hasan anchoring the chase with an unbeaten 66 off 58 balls.
Despite a dramatic finish that saw Bangladesh lose three wickets for just three runs in the final over, Mahedi Hasan struck the winning boundary with two balls to spare.
Bangladesh made a shaky start, losing Saif Hassan in the first over for 4. Tanzid and wicketkeeper-batter Parvez Hossain Emon steadied the innings with a 55-run partnership before Emon fell for 24 off 22 balls.
Captain Towhid Hridoy then combined with Tanzid to add another crucial stand. Hridoy contributed 24 from 19 deliveries, helping Bangladesh move within touching distance of victory.
However, Zimbabwe fought back late as Richard Ngarava, Brad Evans and Sikandar Raza picked up quick wickets to keep the contest alive.
With four runs needed from the final over, Evans dismissed Mosaddek Hossain and Mohammad Saifuddin in successive deliveries, but Mahedi Hasan finished the chase in style with a boundary.
Earlier, Bangladesh's disciplined bowling attack restricted Zimbabwe to 143/7 after opting to field first.
Opener Brian Bennett made 47, while Dion Myers top-scored with 73 off 53 balls, but the hosts failed to capitalize after a promising platform. Bangladesh struck regularly in the second half of the innings and executed two crucial run-outs to deny Zimbabwe a late flourish.
Mohammad Saifuddin was the pick of the bowlers with 2 for 35, while Mahedi Hasan, Rishad Hossain and debutant Abdul Gaffar Saqlain claimed a wicket each.
After disappointing defeats in the Test and ODI series, Bangladesh finished the tour on a high by lifting the T20I series 2-1, denying Zimbabwe a clean sweep of the three formats.
