Bangladesh declared their second innings at 223-8 on the fourth day to set up a 395-run target for the West Indies in the first Test in Chittagong on Saturday.
Skipper Mominul Haque struck 115 and put on 133 with Liton Das (69) to provide the backbone of the Bangladesh innings.
Spinners Jomel Warrican (3-57) and Rahkeem Cornwall (3-87) were the most successful West Indies bowlers.
Bangladesh earlier scored 430 in their first innings before bowling out the West Indies for 259.