Hope, who scored 147 in the first innings, remained unbeaten on 118 in the second innings to take his side to a five-wicket win.
"Really? Thanks for the news", Hope said after being told he had become the first to score a century in both innings of a Headingley game.
Speaking after West Indies sealed their first Test win in England since 2000, Root told Sky Sports: 'They put us under pressure when we batted and then went past us when they got their opportunity to bat.
Brathwaite had also been close to the same slice of history, but his perishing opened the door for Hope to take his place in the record books - his double standing alone in the 534-game history of first-class action at Headingley. "I'm just pleased to get the win".
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.
Substitute fielder Mason Crane took a stunning catch to remove Roston Chase for 30, and the irrepressible Blackwood was stumped chasing a big finish, but Hope and Braithwaite had ensured it was too late for an England rescue act.
Only Australia, with 404 for three back in 1948, had made more to win in the fourth innings of a Headingley Test. Brathwaite's 180-ball innings, including 12 fours, meant the opener had batted for more than 10 hours in the match.
Blackwood, who uppercut Broad for six, tried to win the match in style but, having removed his helmet, he was stumped off Ali for 41, having put on a vital 74 with Shai Hope.
Their partnership allowed England captain Joe Root to declare his side's second innings on 490 for eight, leaving the West Indies, who again harmed their own cause with dropped catches, an imposing target of 322 to win.
This was not just a coming of age but a performance for the ages, the 23-year-old's turnaround from his pained display at Edgbaston reflective of his side's stirring comeback from the ignominy of losing 19 wickets in a day.
"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".
It was yet another drop in this match and although it still yielded a wicket, England would have preferred to see the back of Brathwaite rather than Hope. "(He's got) plenty of courage, plenty of character and plenty of determination".