The West Indies were all out for 117 in their second innings on Sunday, setting Bangladesh a 231-run target to win the second Test.
Bangladesh took seven wickets for 76 runs in little over a session on the fourth day after the West Indies resumed on 41-3.
Taijul Islam finished on 4-36 while Nayeem Hasan took 3-34.
Nkrumah Bonner scored 38 runs for the visitors in the second innings.
The West Indies won the first Test by three wickets to lead the two-match series 1-0.