The hosts carried on from their overnight 95-2 to be bowled out for 265 after 112.5 overs.
Captain Kraigg Brathwaite carried on from his 42 on day one to eventually be dismissed for a marathon 94 off 268 balls including nine fours.
Vice-captain Jermaine Blackwood also showed good form with a 139-ball 63, also including nine boundaries.
Off-spinner Mehidy Hasan Miraz led the way with the ball, with 4-59 off 22.5 overs, while pacers Ebadot Hossain and Khaled Ahmed ended with figures of 2-65 off 28 overs and 2-59 off 22 overs, respectively.
Bangladesh then struggled to 50-2 off 20 overs, trailing the Windies by 112 runs at the close, with Mahmudul Hasan Joy (18) and Najmul Hossain Shanto (8) at the crease. Alzarri Joseph took both wickets for the West Indies.
Chasing an unlikely victory target of 461 and resuming from their overnight score of 47-0, Bangladesh batted through the final day to close on 306-4.
Hasan Joy, who was on 28 overnight shared in an opening stand of 93 with Sakib Hasan. The partnership was broken when Hasan, 14 overnight, was dismissed by Kevin Sinclair for 43. Sinclair dismissed Monimul Haque for just five shortly thereafter but Hasan Joy continued to resist, sharing in a third-wicket partnership of 53 with Saif Hasan, who was eventually dismissed by Akeem Jordan for 38.
Yasir Ali joined Hasan Joy at the crease and together they made the West Indies ‘A’ bowlers toil for just over 29 overs during which they added 117 runs for the fourth-wicket and effectively ensured a draw. Sinclair picked up his third wicket of the innings when he dismissed Ali for 67.
Hasan Joy continued on to 114 not out as he and Shahadat Hossain, who was unbeaten on 20 at the end, took the score past 300 and take the hosts to safety.
Sinclair ended with figures of 3-94. Jordan took 1-37 in the stalemate.
Scores: West Indies 'A' 445 and 220-5 declared; Bangladesh 'A' 205 and 306-4. Match drawn.
Bangladesh carried on from their overnight 50-2 to reach 245 all out off 90.5 overs thanks to Captain Shakib Al Hasan who got his second fifty in the match with 63 and wicketkeeper Nurul Hasan who got 64. Opener Mahmudul Hasan Joy earlier got 42.
Roach, who now has 249 Test wickets, was brilliant for the hosts with 5-53 off 24.5 overs while Alzarri Joseph and Kyle Mayers supported well with 3-55 from 19 overs and 2-30 from 13 overs, respectively.
The hosts, needing 84 to win, got off to a terrible start and were 9-3 after four overs, losing captain Kraigg Brathwaite, Nkrumah Bonner, and Raymon Reifer in quick succession.
Opener John Campbell (28 not out) and vice-captain Jermaine Blackwood (17 not out) then combined to ensure the hosts lost no more wickets, ending the day 49-3 off 15 overs, needing a further 35 runs for a 1-0 lead in the two-match series.
Khaled Ahmed took all three wickets for Bangladesh.
