Set a decent target after Zimbabwe were restricted to 163-7, Bangladesh comfortably reached 166-6 with eight balls in hand to take 1-0 lead in the four-match series.
Zimbabwe made regular breakthroughs but could not stop the run flow as Tamim Iqbal (29), Mushfiqur Rahim (26) and Shakib Al Hasan (20 not out) all made decent contributions to Bangladesh's victory.
Sabbir faced 36 balls and hit four boundaries and a six, taking advantage of a dropped catch by Vusi Sibanda off Sean Williams when he was on 36.
Earlier, Hamilton Masakadza and Sibanda put on a record 101-run opening partnership for Zimbabwe after skipper Elton Chigumbura won the toss and opted to bat first at the Sheikh Abu Naser Stadium in Khulna.
Masakadza hit 79 off 53 balls, equalling his own national record for the highest individual innings, while Sibanda struck 46 off 39.
Bangladesh did well to restrict Zimbabwe in the final few overs, despite Masakadza and Sibanda breaking the team record, set by Sikandar Raza and Chamu Chibhabha's 100-run partnership against Afghanistan in October.
Bangladesh had to wait until the 12th over for their first breakthrough when Sibanda was finally caught at square leg by Tamim Iqbal off Shakib Al Hasan's delivery.

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