England, set a revised target of 246 in 40 overs in a rain-affected match, finished on 246-3 with more than 10 overs remaining.
Jamie Smith launched the run-chase with a dynamic 64, his maiden ODI fifty coming on the opener's Surrey home ground.
Victory meant Harry Brook led England to a clean sweep in his first series as permanent white-ball captain after they had lost their seven previous ODIs, culminating in a woeful exit from this year's Champions Trophy in Pakistan.
Brook was there at the finish with 26 not out.
But it was Jos Buttler, Brook's predecessor as captain, who ended the match in emphatic fashion with a six off fast bowler Jayden Seals that capped his unbeaten 41 off 20 balls.
Earlier, West Indies posted 251-9 off their 40 overs, with recalled all-rounder Sherfane Rutherford top-scoring with 70 and Gudakesh Motie making 63.
Leg-spinner Adil Rashid was the pick of England's bowlers with 3-40 in eight overs.