The hosts took an unassailable 3-0 lead in the series after winning the first two ODIs in Chittagong by 87 runs and 68 runs, respectively.
Anamul and Tamim Iqbal laid the foundations for a big win, putting up an opening stand of 121 runs — their second century partnership in as many matches, which guided Bangladesh to 297-6.
Faced with a massive target in a must-win contest, Zimbabwe were never up to the task and were bowled out for 173 runs in 39.5 overs.
Mashrafe Mortaza (2-24) had an opening-spell blast to leave Zimbabwe 2-22, and spinners did the rest.
Arafat Sunny mopped up the tail to finish with 4-27, his second four-wicket haul in as many matches.
Skipper Elton Chigumbura staged a lone battle for Zimbabwe with an unbeaten 53. Anamul was named man of the match for his 120-ball innings, which included nine fours.
The last two matches of the series will be held at the same ground on Friday and then on Monday.
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