Evin Lewis was adjudged man of the match for scoring an explosive 89 off 36 balls.
Shakib Al Hasan smashed an unbeaten 42-run knock to help Bangladesh post a record total of 211/4.
Some BPL franchises have objected that Steve Smith was selected from outside the draft for the tournament.
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.
Mehidy Hasan's career best 4-29 helped Bangladesh restrict the Windies to 198-8 in 50 overs.
The Windies had won the second ODI of the three-match series after losing the opener.
West Indies registered their first victory to stay alive in the series after suffering a five-wicket loss in the opening match on Sunday.
Bangladesh restricted the Windies to 195-9 and then sailed to 196-5 in 35.1 overs for a comprehensive win.
Bangladesh swept West Indies 2-0 in the recent Test series.
Courtney Walsh, who acts as fast-bowling coach for Bangladesh, said the thumping was bittersweet.
The Windies were clueless against Bangladeshi spin and lost the Test by an inning and 184 runs.
Tamim Iqbal has not played since injuring his wrist during the Asia Cup opener against Sri Lanka in September.
Bangladesh defeated the Windies by an innings and 184 runs to register their first Test series win in four years.
The Windies were reduced to 75 for five at stumps on the the second day, trailing Bangladesh by 433 runs in the first innings.
Bangladesh lead the two-Test series 1-0.
The Bangladesh squad being sent for the Emerging Teams Asia Cup starting December 6 is the first since 2015 to visit Pakistan.
Bangladesh came close to winning series twice in the past two years.
Shakib Al Hasan dismissed Windies opener Kieran Powell on Day 3 to reach the 200-wicket mark in Test cricket.