Chasing a target of 322, the visitors rode of the innings of Hope (120*) and Brathwaite (95) as they registered one of the most thrilling victories in Test cricket.
"We're test cricketers for a reason".
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.
The West Indies, set 322 to win, finished on 322 for five with Hope making 118 not out - his second hundred of the match after his first-innings 147.
SCORES: England 258 all out & 490 for 8 declared. 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.
Hope and Roston Chase kept the scoreboard ticking with a 49-run stand before Chase was removed by Chris Woakes for 30.
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. Alastair Cook dropped Brathwaite on 4 and Stuart Broad dropped him later too which would have exposed Windies' fragile lower-middle order far earlier in the day.
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.
West Indies captain Jason Holder: "I always believed in this group, we've done some pretty decent things in the last few months, it's about putting things together more often than not - I told the guys to believe in and execute our plans".
After getting the kicking we did at Edgbaston, to get the boys back up to achieve what they've done these last five days, it's a great effort - not just by the players themselves but all the support staff.
"He's been going through the form dip but he's looked in good nick".
Brathwaite s 180-ball innings, including 12 fours, meant the opener had batted for more than 10 hours in the match.
Broad missed an opportunity to dismiss Kraigg Brathwaite caught and bowled on his follow through, but the resulting deflection led to the ball hitting the stumps at the non-striker's end, where Hope was out of his crease. "(He's got) plenty of courage, plenty of character and plenty of determination".
"I'm elated, we've been working really hard as a team and we're pleased to get over the line".