Offspinner Rahkeem Cornwall played a key role, ending the match with 9-179 as West Indies clinched the series despite stubborn resistance from Mehidy Hasan.
West Indies reached 41-3 in its second innings at stumps on day three for a lead of 154 runs.
Kyle Mayers became the first batsman to score a fourth-innings double century on debut as West Indies chased down 395 runs for a three-wicket win in the first test.
Mayers, who scored 40 runs in the first innings, ended with 250 runs in the match, the second-highest for a West Indies batsman on debut.
Mominul’s 115 — reaching the 3,000 runs milestone in the process — helped Bangladesh set a fearsome 395 runs target for West Indies.
The West Indies was 75-2 in reply, captain Kraigg Brathwaite on 49 at stumps and newcomer Nkrumah Bonner on 17 in an unbroken 51-run partnership.
Shakib plays his first test since September 2019 following a two-year ban — one year suspended — by the International Cricket Council in October 2019 for failing to report approaches from a bookmaker.
Fourth-placed England's best chance of reaching the final is to win the four-match series against India by a 3-0 or 4-0 margin.
Skipper Tamim Iqbal, Shakib Al Hasan, Mushfiqur Rahim and Mahmudullah all hit half-centuries to help Bangladesh post 297-6
Skipper Tamim Iqbal hit 50 to help the side chase the target in 33.2 overs after the West Indies was bowled out for 148 in 43.4 overs.
All-rounder Shakib was back after being given a two-year ban by the International Cricket Council in October 2019, with one year suspended, for failing to report approaches from a bookmaker.
No primary contact to the leg-spinner has been established as the West Indies touring squad has remained in individual isolation since their arrival.
West Indies are the reigning champions in the T20 World Cup and the only team to have won the tournament more than once.
The Cricket West Indies (CWI) selection panel on Tuesday named the West Indies Test and ODI squads for the upcoming tour of Bangladesh.
Kyle Jamieson took five wickets to leave West Indies struggling at 124/8 on day 2, after New Zealand put a total of 460 in the first innings.
Put into bat, the Black Caps reached 294/6 at Stumps with Nicholls and Kyle Jamieson remaining unbeaten.
Both are set to return to the Caribbean: Roach due to the death of his father, and Dowrich for personal reasons.
A 2-0 series win over the West Indies likely will be enough to lift New Zealand above England into third place on WTC standings.
Cricket West Indies have been awarded for their courage of sending the men's and the women's team to tour England amid the COVID-19 pandemic that had caused a massive disruption to cricket's calendar year.
New Zealand declared their innings on 519-7, while West Indies had added 49 runs without a wicket at the end of day 2.
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