The hosts compiled a decent 277-6 but England raced to victory with 13 balls to spare, playing methodical cricket and ending Bangladesh's unbeaten run in their six bilateral series.
Duckett scored 63, his second half-century in the series, while Billings, who replaced injured Jason Roy in the opening slot, made 62.
Those two knocks kept England on course and Ben Stokes, 47 not out, kept up the momentum to win a series which became heated following a verbal spat between players of the two teams in the second match.
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