Jermaine Blackwood top-scored with 95 as West Indies overcame an early wobble to win by four wickets in the final session of the 5th and final day of the match.
West Indies chased down the victory target of 200-run with four wickets in hand, in their second innings to take an early lead in the three-test series.
Earlier, the tourists had made 318 runs in their first innings. England had scored 204 and 313 all out in the first and second innings respectively.
The second test starts Thursday in Manchester.