Replying to Bangladesh's 229 for 7 in 50 overs, the winning target of Carribbean side was fixed at 194 runs in 35 overs due to late start of the second innings because of drizzling.
Bangladesh skipper Aminul Islam Bulbul played a captain's innings of 52 runs in 65 balls with two fours and later took two wickets for 38 runs in 7 overs to be adjudged man of the match.
The visitors faced an initial batting collapse losing four top order batsmen - Griffith, Campbell, Reifer and Sarwan - for 36 runs in six overs playing amid a drizzle.
However, confident batting by skipper Ian Bishop and other lower order batsmen turned the match to an exciting stage.
The last West Indian pair of Bishop and Reon King needed 17 runs from the last six balls. But Shafiuddin Babu conceded seven runs in four balls before King was run out in the fifth ball. One day debutant Bangladesh took a 1-0 lead beating the touring Caribbean second string by two wickets in the first limited over match at Dhaka. But the visitors levelled 1-1 when the host suffered a 7-wicket defeat in the second one-day match at northern Rajshahi.
This is the first time that Bangladesh won a one-day series against any team from the test playing countries.
Bangladesh earlier notched a sensational one-run victory over Indian A team in the 2nd SARRC Cricket tournament in Dhaka in 1994. The ICC and ACC champions also earned their maiden ODI victory beating Kenya in the three-nation Cricket in India earlier this year.