Brathwaite had made a superb first-innings hundred and was all set for another one as he advanced to 95 just before the tea interval. But as this match has proven, Test cricket is at its most thrilling when it reaches a crescendo.
Jermaine Blackwood then muscled the visitors towards the finishing line with Hope, smashing a typically thunderous 41 off 45 balls including two sixes.
"I ve just been told I was the first man to score a century in both first-class innings at Headingley", Hope said on BBC Radio s Test Match Special. West Indies dominated the third day before Moeen Ali (84) and Chris Woakes (61) forged a 117-run partnership to take England's score to 490.
Stuart Law was a proud Windies coach after his side rallied from their Edgbaston nightmare to beat England in the second Test at Headingley. However, middle-order batsman Shai Hope emerged as a real star for his side as she smashed two consecutive centuries in the Test.
Opener Kraigg Brathwaite had earlier struck a solid 95 after being dropped by Alastair Cook on four.
He breezed to a half-century from 70 balls without ever looking in trouble before playing the match situation after Brathwaite's dismissal. "For us, we have 317 runs to make tomorrow and we've still got to believe that we can win the game", he contended.
This was also, excluding fixtures against Bangladesh and Zimbabwe, just the West Indies fourth Test win away from home in 88 matches spanning 20 years.
Runners Mourn Olympian David Torrence's Death At 31
Torrence was a two-time champion of the Hoka One One Long Island Mile in NY , and was aiming for his third title on September 6. Fellow HOKA athlete Kyle Merber announced they would name the Long Island Mile race he organizes every year after Torrence.
The pitch was turning and the cloud cover gave England's pacemen real hope but young Bajans Brathwaite and Hope negotiated the testing conditions superbly.
Meanwhile West Indies captain Jason Holder insisted he d never lost faith in his players.
"It was nerve-wracking with Jermaine Blackwood, a lot of credit must go to him with how well he played and not changing his style of play".
"At no point were we complacent in the game - I'm really proud of the effort we showed to get in a position to win, but unfortunately it wasn't to be".
Under heavy criticism following a heavy innings defeat inside three days last weekend, the Caribbean side responded strongly by piling up 329 for five in their first innings at the close - a lead of 71 heading into day three today. Shai Hope has joined him in the crease.
It all represented a testament to England's batting strength in depth after they started on Monday just two runs in front and with three wickets already down. "(He's got) plenty of courage, plenty of character and plenty of determination".
"He's been going through the form dip but he's looked in good nick".