Pollard and Ramdin compiled 145 runs for the 6th wicket to pull the homeside from the jaws of defeat following a top order batting collapse which left them reeling on 34 for five.
Pollard started slowly but accelerated his innings later to finish with a top score of 89 while Ramdin scored 74, starting quickly but curbing his enthusiasm later in the innings.
After their impressive counter attack the two batsmen perished in the dying stages of the chase to give Bangladesh a chance to make a come-back.
Jason Holder hit three fours to reach 22 and kept his nerve with Sunil Narine to ensure West Indies take a one-nil lead in the three-match ODI series.
Al-Amin Hossain, who finished with 4-51, snatched three early wickets as the Caribbean side slumped to 20-3, chasing the visitors’ 217 for nine.
All the top order batsmen were dismissed cheaply, Chris Gayle scoring 3, Kirk Edwards 10, Darren Bravo 7 and Lendl Simmons 0.
But the efforts of the two Trinidadians were vital in guiding West Indies to their win by forcing Bangladesh to concede a match winning position.
Pollard struck five fours and six sixes facing 70 balls while Ramdin, who faced 76 balls, accounted for six fours and a six to transform the match and carry West Indies to the brink of victory before the partnership expired.
A brief rain delay may have interfered with the flow of the partnership with Ramdin departing shortly after, playing on to the base of his off stump.
Less than three overs later Pollard’s whirlwind innings ended as he mistimed a shot aimed for over the long on boundary and was caught.
Earlier, Bangladesh opener Anamul Haque reached his third ODI century to guide his team to a competitive total.
His knock came off 133 balls with nine fours and one six after West Indies captain Dwayne Bravo won the toss and asked Bangladesh to bat first.
Haque held the innings together, but was alone as wickets fell regularly with West Indies skipper Dwayne Bravo taking four for 32 runs in his seven overs.
He and Nasir Hossain scraped together 53 runs for the sixth wicket, Hossain contributing 22, the same score as opener Tamim Iqbal.
The second ODI between West Indies and Bangladesh is also scheduled for the national stadium in Grenada on Friday.
Bangladesh Innings: 217/9 (50 Overs). Anamul Haque 109, Tamim Iqbal 26, Nasir Hossain 26, Mushfiqur Rahim*+ 12; Dwayne Bravo 7-0-32-4, Jason Holder 9-0-39-1, Chris Gayle 4-0-19-1, Ravi Rampaul 10-1-48-1.
West Indies Innings: 219/7 (39.4 Overs). Kieron Pollard 89, Denesh Ramdin+ 74, Jason Holder 22, Kirk Edwards 10; Al-Amin Hossain 8.4-0-51-4, Mashrafe Mortaza 8-2-22-1, Mahmudullah 4-0-20-1, Sohag Gazi 9-1-46-1.