West Indies reached 228-6 with three overs remaining at Shere Bangla National Stadium after Bangladesh was sent in to bat at here Bangla National Stadium and bowled out for 227 in the final over.
Samuels' fourth ODI century included 17 boundaries and two sixes, and he smashed 24 runs in the 45th over from medium-pacer Rubel Hossain to ensure victory.
Kieron Powell scored 47 to offer support for Samuels after opener Chris Gayle had been removed for 4 to prolong his poor run of form.
Samuels also perished to Mashrafee Bin Mortaza, caught by Mahmudullah near the boundary.
In Bangladesh's innings, Mahmudullah scored 52 and Mushfiqur Rahim made 38 as the home side struggled against a disciplined bowling performance by the visitors.
Narine claimed 4-37 from 10 overs to restrict Bangladesh's scoring, while left-arm spinner Veerasammy Permaul took 2-40 and Darren Sammy captured 2-46.
Bangladesh recorded its biggest ever ODI win last weekend, beating West Indies by 160 runs.
Narine removed openers Tamim Iqbal for 22 and Anamul Haque for 33. He also claimed the wickets of Naeem Islam for 4 and Mashrafe bin Mortaza for a duck.
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