After the tourists were limited to 129 for seven, medium-pacer Taskin claimed two early wickets and then returned to seal victory in the 19th over when he dismissed all-rounder Akeal Hosein, who fell for 31 to a catch at backward-point just as he was threatening to win the match off his own bat as the West Indies were folded for 102.
Taskin finished with three for 16 while fellow seamer Tanzim Hasan Sakib and wrist-spinner Rishad Hossain took two wickets each.
Hosein and Roston Chase rescued the Caribbean team from an abject collapse at 42 for six in the ninth over, putting on 47 for the seventh wicket before Hossain struck two critical blows when he bowled Chase for a topscore of 32 and then had Gudakesh Motie taken at long-off first ball to tilt the balance of the match decisively Bangladesh’s way.
Earlier, they were again indebted to another late flurry from Shamim Hossain in getting up to their final total.
He belted two fours and two sixes in getting to an unbeaten 35 off just 17 balls, dominating an unbroken 41-run eighth-wicket partnership with Tanzim (9 not out). Shamim's efforts earned him the "Man of the Match" award.
Like the West Indies, Bangladesh also got off to a bad start, slumping to 52 for five by the 11th over, the only contribution of note at that point coming from Mehidy Hasan Miraz with 26.
However, Jaker Ali's 21 avoided a complete collapse and his demise opened the way for Shamim to dominate the end of the innings with Tanzim offering sturdy support.
Motie's left-arm spinners earned him the best figures for the West Indies with two for 25.
Swept 3-0 in the preceding One-Day International series after the teams shared the Tests 1-1, Bangladesh will now be seeking their own 3-0 sweep when the final match is played at the same venue on Thursday.
