West Indies, the 1975 and 1979 World Cup champions had to win this five-match series 5-0, or 4-0 with a tie or no result, to avoid the indignity of going through qualifying for the next edition in England in two years' time.
But Bairstow's 100 not out saw England easily chase down a meagre target of 205 in a match reduced to 42 overs per side by a wet outfield as the hosts went 1-0 up with 67 balls to spare.
Ben Stokes returned figures of three for 43 with only Windies captain Jason Holder's 41 not out taking the tourists Stokes ended the match in style with a six off spinner Ashley Nurse.
Brief scores
West Indies 204-9, 42 overs (B Stokes 3-43)
England 210-3, 30.
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