The visitors reached the 170-run target in 35.1 overs with only two wickets down. Brandon King finished unbeaten on 81, while John Campbell scored 49 and Kavem Hodge made 31 not out.
Bangladesh, resuming on 79/6, were bowled out for 142 in their second innings. The unresolved position at the end of day three has therefore become a final result. The second Test is due to begin in Sylhet on 5 November.