West Indies entered the day in a strong position with England on 171 for three - only two runs ahead - but let the advantage slip, allowing the hosts to pile up 490 for eight declared in their second innings at Headingley.
Hope, who became the first batsman in history to score two centuries in each innings of a first-class game at Headingley, was brilliantly supported by opening batsman Kraigg Brathwaite, who scored a magnificent 95.
"Shai Hope was just outstanding, to come out in the first innings and score his maiden Test century". If West Indies had been set a ridiculously high target they would have shut up shop for a draw.
Former England captain Michael Vaughan will not be the only one to eat a slice following the Windies' shock win.
Victory gave Windies their first Test match win in England since 2000, and just their fourth Test win away from home in 88 matches spanning 20 years, excluding matches against Bangladesh and Zimbabwe. We fought really hard, we knew we needed this win to keep alive the series.
Moeen Ali was operating from the Kirkstall Lane End.
As the day wore on, Alastair Cook's early miss, dropping Brathwaite at first slip off Broad with the Windies on just 11 without loss, had looked ever more costly.
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Stuart Broad had a hand in both wickets to fall on the final morning at Headingley - literally in the case of a freaky run out of Kyle Hope - but Kraigg Brathwaite escaped two dropped catches to continue his outstanding match.
Chase benefited from a Ben Stokes overthrow to the tune of five runs, but fell for 30 when substitute fielder Mason Crane claimed a great diving catch at mid-on.
Stuart Law was a proud Windies coach after his side rallied from their Edgbaston nightmare to beat England in the second Test at Headingley.
The opener pressed on to register his fifty off 98 balls with seven fours, Shai Hope reaching the same milestone in 28 few balls as England's frustration grew.
Blackwood, who uppercut Broad for six over third man, tried to win the match in the grand manner.
Even though Blackwood was stumped by Jonny Bairstow off the bowling of Moeen, Hope hit the winning runs the very next over and finished unbeaten on 118 off 211 deliveries, which included 14 boundaries.
It proved the one that got away as Brathwaite rode his luck to contribute a further match-shaping 91 runs in a crucial stand with Hope, who was also grassed by Cook after he had passed his hundred, the final nail in England's coffin.
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