Bangladesh posted a total of 270 all out in the first innings after opting to bat first at the toss, with opener Shadman Islam hitting 105 but Windies spinner Jomel Warrican took figures of 5-53 in 23.4 overs.
The Caribbean then replied with a total of 245 to concede a led 30, with John Campbell and Tagenarine Chanderpaul scoring fifties but Bangladesh spinner Nayeem Hasan (4-33) initiated a collapse.
The game turned in the third innings when visiting captain Roston Chase took 5-36 in 15.4 overs to roll the hosts out for just 142 and set up up the run-chase of under 200.
Campbell and Brandon King then put up and opening partnership of 112 to settle the Windies dressing room but the latter eventually fell one run short of his fifty.
Chanderpaul was also out cheaply but King (81 not out) and Kavem Hodge (31) put on an unbroken partnership of 50 runs to take the West Indies over the line and a series lead.
West Indies captain Roston Chase lauded his Test team for "going in the right direction as a unit" after going the entire calendar year without losing a Test series following their win in Bangladesh.
Towhid Hridoy struck Bangladesh's highest-ever individual score in Test cricket in the second match against West Indies in Sylhet, but the tourists got the draw required to seal a series victory.
Bangladesh have recalled seamer Taskin Ahmed into the squad ahead of the first Test against West Indies that starts in Mirpur on Wednesday.
Darren Sammy believes West Indies have taken a significant step forward after holding India to a 2-2 T20I series draw, with the head coach encouraged by the return of his team's confidence and trademark swagger.
