Mushfiqur (44) and Mahmudullah (48) came together with the home side on a vulnerable 30 for three, but their profitable fourth-wicket partnership helped push Bangladesh towards their target of 218.
Nasir Hossain then contributed an unbeaten 39 to wrap up the series for the hosts, who had earlier bowled the Windies out for 217 despite a blistering 85 from Keiron Pollard.
Asked to bat first, opener Kieran Powell was the first wicket to fall for 11, beaten by a Sohag Gazi delivery and stumped by Mushfiqur. Marlon Samuels (one) quickly followed, run out by a direct hit from Mahmudullah, and Chris Gayle's dismissal two overs later to become Shafiul Islam's first victim left West Indies on a precarious 17 for three after just seven overs.
However, Darren Bravo and Pollard then joined forces to dig the visitors out of a hole. Pollard raced to his fifty in just 49 balls, with six sixes and two fours, and the century partnership soon followed.
The pair had put on 132 when Pollard was eventually dismissed for 85, clean bowled by Mominul Haque. Darren Sammy barely troubled the scorers when he became Mominul's second wicket but Bravo continued to defy the Tigers and passed his half-century soon after, with three boundaries and one six in his 102-ball knock.
However, Bravo's dismissal on 51 - Mominul taking the catch at short cover from Mahmudullah's delivery - signalled the beginning of the end for the Windies and Shafiul claimed two more wickets to help set a modest target for the home side.
Their reply began in faltering fashion with Tamim Iqbal (eight), Anamul Haque (nought) and Jahural Islam (10) all falling to Kemar Roach, but Mushfiqur and Mahmudullah smashed 14 boundaries between them to wrestle the initiative away from West Indies.
Mahmudullah eventually fell just two runs short of his fifty to a teasing delivery from Sunil Narine, and the same bowler struck again four overs later to end Mushfiqur's spell at the crease. Mominul further frustrated the Windies attack, though, with a knock of 25 before becoming Narine's third victim.
And when Roach dismissed Sohag and Abdur Razzak to complete a five-wicket haul, the home supporters were becoming a little fidgety. However, Nasir's stubborn innings saw Bangladesh home, the right-hander hitting the winning runs at the end of the 44th over to spark celebrations.