After losing four wickets for 107 runs before lunch the test could have been over on the fourth day but captain Mushfiqur Rahim's unbeaten 70 and Mahmudullah's 66 kept Bangladesh's fight alive.
Their 130-run partnership lasted 50.5 overs and frustrated a West Indies attack buoyed by their early successes.
Kemar Roach had Shamsur Rahman caught behind for four before Imrul Kayes (25) lobbed a gentle catch to cover off Chris Gayle's off-spin.
Sulieman Benn, who took five wickets as Bangladesh were bowled out for 182 in the first innings, removed Mominul Haque for 12 with wicketkeeper Denesh Ramdin snaffling up an inside edge.
Tamim Iqbal made 53 off 107 balls before he was bowled by a sharply turning delivery from Benn.
The partnership between Mushfiqur and Mahmudullah did not provide gripping viewing but it restored pride for the tourists.
Mahmudullah, however, eventually went when he attempted to drive a Roach delivery outside the off stump that moved into him and he nicked an inside edge that was superbly taken by a diving Ramdin.
(Reporting by Simon Evans, editing by Tony Jimenez)
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