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With England set a modest target of 205 in a match reduced to 42 overs per side because of a weather-delayed start, the outcome was never really in doubt.
The main question was if Bairstow had time to reach three figures for the first time in ODIs.
Bairstow got there in the 31st over and a few balls later Ben Stokes sealed the win by launching Ashley Nurse over the boundary rope.
West Indies' defeat means they will have to qualify for the 2019 World Cup.
They made a fast start to their innings with Chris Gayle, dropped by Joe Root in the first over, thumping three huge sixes as the visitors reached 45 without loss after five overs.
Root made amends when he sprinted full steam to catch Gayle on the boundary for 37 and while Shai Hope (35), Marlon Samuels (17) and Jason Mohammed (18) all made starts, wickets fell at regular intervals with Stokes the main dangerman.
Hope was superbly caught one-handed by man-of-the-match Bairstow on the mid-wicket boundary off of the bowling of Stokes, who finished with figures of three for 43.
Jason Holder's 41 not out helped West Indies past the 200-mark but it never looked like being enough.
Alex Hales was caught off the bowling of Jerome Taylor for 19 but Root joined Bairstow for a second-wicket stand of 125 before Root was bowled by Kesrick Williams for 54.
Williams also removed captain Eoin Morgan for 10 but Stokes joined Bairstow as England cantered home under the lights.
The second match is at Trent Bridge on Thursday.