Zimbabwe won the toss and sent Bangladesh to bat first. Blessing Muzarabani removed Tamim Iqbal for a duck in the third over with no run on the board.
Off the first ball of the third over, Tamim went to cut but managed only an edge to the wicketkeeper.
Shakib Al Hasan, Mohammad Mithun and Mosaddek departed after scoring 19, 19 and five runs.
Muzarabani has bagged two wickets and Tendai Chatara and Richard Ngarava took one each.