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Irresponsible batting by the Bangladeshi top-order saw them all out for 182 runs in 48 overs after the visitors scored 266 for four in 50 overs.
Bangladesh started disastrously and lost six wickets before reaching 100 with only Mohammad Ashraful, the world's youngest Test centurion, providing any resistance with a career best 44.
Ashraful was dropped from the team during Bangladesh's South Africa tour in October this year.
Drakes, who captured four wickets for 26 in the rain-abandoned match in Chittagong, grabbed a career-best four for 18 after he ripped the top off the Bangladeshi top order by grabbing the first three wickets to fall.
Bangladesh captain Khaled Mashud and Naimur Rahman proved some late-order resistance, adding 71 runs off 95 balls in their seventh wicket stand to help Bangladesh reach their face-saving score.
Mashud made 41 before he was bowled by Cameron Cuffy, who ended up with two for 33, while Rahman made 37 before he was run-out.
Earler, a magnificent third wicket stand of 146 by Marlon Samuels and Ramnaresh Sarwan helped West Indies to their total.
Sarwan's unbeaten 102 also helped him bring up 1 000 one-day international runs. He reached his century when he smashed the last ball of the innings to the rope off Tapas Baishya.
Sarwan, who was dropped by Ehsanul Haque at short extra cover off Manjurul Islam, hit 11 fours off 100 balls to take his one-day tally to 1 044 in 28 matches.
West Indies were 37 for two in the 13th over before Sarwan and Samuels began their partnership.
The pair, who worked the ball around and slowly built their innings, exploded in the 30th over and added 82 runs in nine overs to lift the score to 183 for three in 39 overs when Samuels was bowled by Naimur Rahman for 82.
He hit seven fours and two sixes off 119 balls and also survived a run-out after being obstructed by bowler Mohammad Rafique on 29 in the 27th over.
The third and final one-day match will be played in Dhaka today.
C.Gayle c K.Mashud b M.Islam 21
W.Hinds lbw b Tapas Baishya 0
M.Samuels b N.Rahman 82
R.Sarwan not out 102
R.Powell b Ehsanul Haque 10
D.Ganga not out 21
Extras: (b-5, lb-20, nb-1, w-4) 30
Total: (for four wickets, 50 overs) 266
Fall of wickets: 1-0, 2-37, 3-183, 4-214.
Bowling: Manjural Islam 10-3-20-1, Tapas Baishya 10-1-42-1, Ehsanul Haque 10-0-57-1, Mohammad Rafique 10-2-41-0, Naimur Rahman 7-0-53-1, Alok Kapali 3-0-28-0
Al Shahriar b Drakes 0
Ehsanul Haque c Gayle b Drakes 9
M.Ashraful c Hinds b Nagamootoo 44
H.Bashar b Drakes 0
S.Hossain lbw b Lawson 13
A.Kapali stp. Jacobs b Nagamootoo 12
K.Mashud b Cuffy 41
N.Rahman run-out 37
M.Rafique c Sarwan b Drakes 0
Tapas Baisya not out 2
M.Islam b Cuffy 0
Extras: (lb-7, nb-3, w-14) 24
Total: (all out, 48 overs) 182
Fall of wickets: 1-0, 2-34, 3-34, 4-59, 5-84, 6-94, 7-165, 8-179, 9-182.
Bowling: Drakes 9-4-18-4, Lawson 10-1-42-1, Cuffy 9-0-33-2, Nagamootoo 10-2-32-2, Gayle 8-1-38-0, Samuels 2-0-12-0.