England finished on 294-1 after Joe Root smashed a six off Marlon Samuels to win the series 4-0 with the second ODI a no-result due to rain.
Bairstow, who also got a ton in the first match, hit 17 boundaries in his 114-ball innings. Jason Roy was the only English wicket to fall, trapped leg before wicket for 96 by Miguel Cummins in the 22nd over, after an opening stand with Bairstow of 156.
Earlier, Shai Hope struck 72 and Ashley Nurse added a late-innings 31 not out off 12 balls to help West Indies score 288-6 after England won the toss.
Chris Gayle hit five sixes, including four in a row across two overs, in a 29-ball 40 at the top of the innings to get the tourists off to a quick start.
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Hope's knock was more measured, coming off 95 balls and including four fours and a six.
All eight of West Indies' batsmen got into double figures as they looked to avoid a series whitewash.