With the West Indies in cruise control, England were beginning to worry as the tourists were edging ever closer to victory.
Anderson came within an inch of claiming England's second wicket only for Brathwaite's edge to fall short of Westley at third slip but soon did bag another in the most fortunate of circumstances.
"I woke up this morning and sent a text in our team chat and told the guys to believe, just go and believe and execute our plans".
Brathwaite and Hope, who both starred in the West Indies' first innings, steadied the ship with a 145-run partnership, during which both batsmen surpassed their half-centuries. It was the Windies' first in a test in England for 17 years. "I'm pleased he got some runs and made the game easier for me".
Hope top-scored with 147 runs in the Windies' first-innings 427, putting on 246 for the fourth wicket with fellow centurion Kraigg Brathwaite.
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With England having set a formidable target - the Windies started the day needing 317 with all wickets intact - on a turning pitch and under cloud cover, the two young batsmen negotiated their testing examinations superbly.
The home side did not bowl as consistently as they might have wished with Chris Woakes, Stokes and Moeen Ali proving expensive at times and aside from the first hour, England's bowlers found less from a fifth-day pitch than they might have expected.
Shai Hope s single off Broad completed a superb century off 175 balls, including 13 fours, as he became the first West Indian to score hundreds in both innings of a Test in England since Gordon Greenidge at Old Trafford in 1976.
"[He's got] plenty of courage, plenty of character and plenty of determination, and he played really well".
He then survived a potential stumping as Blackwood, who was put down by Stokes in the deep, took off his helmet in preparation for scoring the winning runs only to be stumped off Moeen with two required. The list of dropped chances was added to when Brathwaite offered a return catch to Broad which he couldn't hold, but the ball deflected into the non-striker's stumps with Hope well out of his ground.
Watch live coverage of the third Test between England and Windies on Sky Sports Cricket with over-by-over commentary and in-play video clips on our live match day blog, available across Sky Sports digital platforms, from 10am on Thursday, September 7.