After limiting England to 258 all out in their first innings - Kemar Roach and Shannon Gabriel taking four wickets apiece - West Indies kept their noses in front for the most part and a decisive 144-run partnership between Hope and Kraigg Brathwaite on the final day followed their 246-run union in the first innings.
"Shai Hope was just outstanding, to come out in the first innings and score his maiden Test century".
It was all the more impressive given that just over a week earlier they d suffered an innings and 209-run defeat inside three days in the first Test at Edgbaston.
Root said: "You've got to take your chances in Test cricket".
Meanwhile West Indies captain Jason Holder insisted he d never lost faith in his players.
"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".
Root had opted to end England's innings with two wickets in hand on the fourth evening in an effort to make inroads into the Windies batting line-up but ultimately his decision allowed the tourists a route back into the contest. But after resuming on five without loss, Windies scored more than 300 runs on the last day at Root's Yorkshire home ground.
Hope, whose average of 18 over his previous 11 tests had never reflected his potential, said he trusted the win would be a stepping stone towards a brighter future for the team. We fought really hard, we knew we needed this win to keep alive the series.
But Broad did reduce the West Indies from 46 for none to 53 for 2.
A mixture of calm defiance combined with judicious attacking frustrated the England bowlers to the point that James Anderson, who had a run-in with opener Kieran Powell, and Stuart Broad, who was warned for kicking the pitch petulantly, became increasingly disgruntled.
In 534 first-class matches at Headingley, no batsman had scored a century in both innings - but though Brathwaite missed out, caught at slip off Moeen Ali just before tea for 95, Hope battled on to reach the landmark in 276 minutes.
Brathwaite again made the best of his good fortune to mix caution with admirable aggression as he faced 180 balls, striking 12 fours.
Hope, the first West Indies batsman to score hundreds in both innings of a Test in England since Gordon Greenidge at Old Trafford in 1976, hit the winning runs with two off Chris Woakes as the West Indies won with 28 balls to spare.