Set a revised target of 342 in 43 overs under the DLS method, Zimbabwe were bowled out for 218 in 37.2 overs.
Mohammad Saifuddin claimed 4-41 for Bangladesh, with Sikandar Raza making 61.
The two teams will now play in a two-match Twenty20 international series, with the matches scheduled at Dhaka's Sher-e-Bangla National Stadium on March 9 and 11.