Set 261 to win, the home side slumped to 215 all out.
Top scorer was Chaminurwa Chibhabha, who contributed 57, while Vusi Sibanda made 46, but their partnership of 72 from 132 balls was made at a painfully slow rate and put the home side well behind the required rate.
Chibhabha made his runs off 116 deliveries, fewer than a run every two balls; Sibanda needed 80 Sunday.
To make matters worse, the duo failed to find the boundary in almost 28 overs.
Hopes were raised with eight overs left when Elton Chigumbura went into bat, with Zimbabwe needing well over 10 an over.
He hit the Bangladeshi attack for four sixes but was bowled for 47 after facing just 26 balls.
It was captain Habibul Bashar who set up the Bangladesh innings in the first place with 78 off 87 balls and by Aftab al Hasan with 68.
They put on 105 for the fourth wicket, enabling Bangladesh to post a good score and as it turned out the foundation for victory.