Shakib smacked 42 off 26 balls to help Bangladesh post 211/4 before taking his maiden five-wicket haul to script a 36-run victory
Shakib now has two demerit points, having been handed one for an offence during a Twenty20 against Sri Lanka in March this year.
West Indies raced to 130-2 in just 10.5 overs after Cottrell and his fellow pacers bowled out Bangladesh for 129 runs in Sylhet.
The first T20 will be played on Monday at the Sylhet International Cricket Stadium, while Dhaka will host the next two matches on December 20 and 22
Tamim Iqbal and Soumya Sarkar slammed half-centuries as Bangladesh cruised to an eight-wicket win over the West Indies
Saifuddin and Mithun replaced Mosaddek Hossain and Sabbir Rahman, who played in Bangladesh's last T20 series against the West Indies in July-August
The Caribbean side last won an ODI series in 2014 at home against Bangladesh and are desperate to end the drought
Hope smashed a brilliant century as West Indies edged Bangladesh by four wickets in the second one-day international
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