Captain Roston Chase claimed 3-25 to put West Indies in a strong position in Mirpur.
Bangladesh slumped to 79-6 in their second innings, holding a lead of 106 runs.
Jomel Warrican supported Chase with figures of 2-19 after a five-wicket first innings haul.
Nayeem Hasan took 4-33 to bowl West Indies out for 245 earlier in the day.
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