Chris Gayle then scored an unbeaten 110 to help West Indies to 262 for five when bad light stopped play shortly before the scheduled close.
Resuming on their overnight score of 406 for nine, the tourists lost Mohammad Rafique for 111 to be bowled out in 135.3 overs.
Captain Habibul Bashar top-scored with 113, after leading the way on Friday, while fast bowler Pedro Collins, with four for 83, was chief wicket-taker for West Indies.
Bangladesh's previous Test best was 400, achieved against India in Dhaka in 2000-01.
West Indies lost Devon Smith run out for nought with two on the board but Ramnaresh Sarwan 40, Brian Lara 53 and Dwayne Smith 42 restored confidence.
The Bangladeshis have yet to win a Test after 28 matches, losing 26 Tests and drawing two -- both against Zimbabwe.
Bangladesh was granted Test status in 2000.
