Brathwaite went on to make a further 91 runs and once again dovetailed superbly with Hope in a 144-run stand - following their 246-run partnership in the first innings - that took a large chunk out of the Windies' victory target.
This match alone will not save Test cricket from declining interest and its struggle for position against cricket's other formats.
Bowling coach, Roddy Estwick, conceded Monday that West Indies had failed to properly negotiate the big moments on the penultimate day of the second Test but warned pundits not to write off the Caribbean side ahead of a tense final day on Tuesday. But Jermaine Blackwood's response, with the Headingley floodlights shining bright on a gloomy day, was to loft Anderson's second delivery with the new ball for a stunning straight six.
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.
Having successfully steered his team through six nervy overs before stumps yesterday, Brathwaite was relieved to see an edge off Stuart Broad put down by Cook in the slips. "For us, we have 317 runs to make tomorrow and we've still got to believe that we can win the game", he contended.
If England secure the win, they would already have the Wisden Trophy in the bag ahead of the third and final Test at Lord's.
Law also had words of praise for the 24-year-old Brathwaite, saying: "He's not the most fashionable-looking cricketer but he gets the job done and he's one of my first picks every game".
The ball cannoned off Broad's thigh to run out Hope's brother Kyle for a duck at the non-striker's end but, given what followed, England will have preferred Brathwaite to be the man walking off.
"Shai Hope played exceptionally well", said Root. I'm extremely proud of these guys.
In Kraigg Brathwaite and Hope, they have two batsmen of real Test match pedigree who they can build around for the next decade.
"It's put a lot of doubters out there to bed - it takes a lot of character to come out after the beating we got to beat the same side, especially in seam friendly conditions".
"I'm elated, we've been working really hard as a team and we're pleased to get over the line".
"I'm really proud of Shai Hope, he's been going through a form dip but hopefully the pretty 20s are gone and the pretty 120s are here to stay".