Liton steadied Bangladesh innings building a 93-run fifth wicket stand with Mahmudullah Riyad after Bangladesh lost their first 4 wickets for only 74.
After Liton departed while Bangladesh at 207, Afif Hossain and Mehidy Hasan Miraz added valuable 58-run stand in the seventh wicket to post a fighting scored on the board.
Afif scored 45 off 35 and Mehidy 26 off 25 while Mahmudullah scored 33.
Earlier, 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 cheaply.
Luke Jongwe scalped three wickets, Muzarabani and Richard Ngarava took two each.