Chasing a token target of 28 in their second innings, West Indies reached it without loss, captain Kraigg Brathwaite hitting the winning runs to seal a 1-0 series triumph.
It was a stunning result for the home team after being outplayed in the first two drawn tests, and ensured another barren England trip to the Caribbean.
England have only one series win in the West Indies in the past half-century, a 3-0 victory in 2003-04.
With the ecstatic home fans dancing in the stands, the England players shook hands with Brathwaite and John Campbell.
Even before the England team trudged off the field in dismay, however, commentators were calling for Joe Root to step down as captain.
Although one of the world's best batters, with a test average of nearly 50, Root has rarely looked a natural leader and has earned a reputation as an unimaginative captain.
The first two tests were drawn on flat pitches in Antigua and Barbados.