West Indies 265 (Brathwaite 94, Blackwood 63, Mehidy 4-65) and 88 for 3 (Campbell 58*, Blackwood 26*, Khaled 3-27) beat Bangladesh 103 and 245 (Nurul 64, Roach 5-53, Joseph 3-55) by seven wickets
West Indies finished on 88 for 3 after resuming the fourth day on 49 for 3 in the second innings, needing 35 more runs to clinch the series opener.
Out-of-form Bangladesh had given the hosts a scare on Saturday, at one stage reducing the West Indies to 9 for 3 at the Sir Vivian Richards Stadium.
Bangladesh medium pacer Khaled Ahmed, who took three wickets in 11 balls on day three, was unable to break through again. He finished with 3 for 27 from eight overs.
The second and final Test starts Friday in St Lucia.