Elton Chigumbura led the way with a vital fifty as Zimbabwe defeated hosts Bangladesh by 38 runs in the first match of the three-team tournament.
Chigumbura hit a patient 64 to steer the tourists to 205-9, while Shakib Al Hasan claimed 3-23 and Naeem Islam took 3-32 for the hosts.
The Tigers slumped to 33-4 before Shakib scored a defiant 52 but the hosts were dismissed for 167.
Bangladesh now have to beat Sri Lanka to stay in contention for a final spot.
Chigumbura's 64 came off 95 deliveries and contained five boundaries, while captain Prosper Utseya claimed 2-22 for the tourists.The second match of the series between Sri Lanka and Zimbabwe takes place in Mirpur on Monday.