Bangladesh, who were forced to follow-on, resumed on 256 for five but lost their last five wickets for 58 runs to leave the hosts needing only 13 runs to win.
Quick bowler Kemar Roach ended with four for 64 and medium pacer Shannon Gabriel picked up two wickets as West Indies cleaned up the tail.
Bangladesh captain Mushfiqur Rahim, who helped his team avoid an innings defeat with a determined 116, his third test century, was the last man to go when he holed out to Darren Bravo at deep mid-wicket off Jerome Taylor.
Openers Chris Gayle and Kraigg Brathwaite then took 16 balls to reach the target and give West Indies a 1-0 lead in the two-match series.
The second test in St Lucia starts on Sept. 13. (Reuters)