Tamim struck 84 off 93 balls at the Sher-e-Bangla National Stadium as Bangladesh cruised to 171-2 in just 28.3 overs.
The hosts earlier dispatched Zimbabwe for 170 in 49 overs as left-arm spinner Shakib Al Hasan claimed 3-43.
Sikandar Raza top-scored for the visitors with 52 as well as taking 2-53.
Bangladesh never looked in trouble, with Shakib striking twice in the first over after skipper Mashrafe Mortaza won the toss and opted to bowl.
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Peter Moor supported Raza with 33 before Rubel Hossain (2-24) took two wickets off successive balls, fast-tracking Zimbabwe's collapse and becoming Bangladesh's fifth bowler to claim 100 ODI wickets.
Mustafizur, who claimed 2-29, produced an off-cutter to uproot Blessing Muzarabani's off-stump and end Zimbabwe's innings.
Tamim, who hit eight fours and a six, lost his opening partner Anamul Haque early on for 19 but added 78 with Shakib (37) for the second wicket to set up the innings for Bangladesh.
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