Jaker Ali's Test-best innings of 91 lifted Bangladesh to 268 all out in their second innings and left the West Indies with a daunting victory target on Tuesday's fourth day of the second and final Test at Sabina Park.
Having trailed on first innings by 18 runs, the home side are now required to go against history in pursuit of 287 for a 2-0 sweep of the series.
The highest successful run-chase in a Test at this venue is 212 by the West Indies against Sri Lanka in 2003.
Ali's belligerent innings, which occupied 106 deliveries and was embellished by eight fours and five sixes, has tilted the balance heavily in favour of Bangladesh levelling the series and earning their first Test victory in the Caribbean since a 2-0 sweep of a depleted home team in 2009.
Such was the extent of the 27-year-old right-hander's dominance of the morning that he contributed 62 of the 75 runs added by the tourists after resuming at the overnight position of 193 for five.
Alzarri Joseph and Kemar Roach took three wickets each for the West Indies, Joseph ending Ali's quest for a maiden Test hundred when he was last out, caught at deep midwicket half an hour before lunch.