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
John Campbell hit the winning six as West Indies needed only seven overs on the fourth day to beat Bangladesh by seven wickets in the first Test in Antigua.
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.
Campbell accelerated his scoring rate on Sunday and finished the match with a six over long-off against Najmul Hossain Shanto. He hit nine fours and one six in his unbeaten 58. Jermaine Blackwood at the other end was on 26 not out after the pair came to their team's rescue with a fourth-wicket partnership of 79.
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.
West Indies scored 265 in the first innings. Bangladesh had 103 and 245. Kemar Roach was named Player of the Match for this match haul of 6 for 74.
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.