Set a challenging 322 for victory, the Windies finished the day on five without loss.
Hope, who struck stylish boundaries on both sides of the wicket, was 118 not out following his first-innings 147.
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.
It is quite the turnaround from the Windies following an innings-and-209-run trouncing at Edgbaston - a defeat that left many former players bemoaning the current set-up - but they have silenced their critics in the most emphatic fashion, having bossed the majority of this Test.More news: Microsoft exec enthuses about "no compromise" Xbox One X
In Kraigg Brathwaite and Hope, they have two batsmen of real Test match pedigree who they can build around for the next decade. "I'm just pleased to get the win".
"We're test cricketers for a reason". Windies still required 76 runs to win and just over 20 overs in which to do it but the tourists had clearly made a decision to go for it.
There seemed little danger of a Windies win when Joe Root declared his side's second innings on 490 for 8 late on Monday'sfourth day.
After being in control for the first three days, West Indies is now up against it and needs the second-highest total to win a test at Headingley, and the highest since Don Bradman's Australia team made 404-3 batting last in 1948.More news: Clear Link Between Heavy Vitamin B Intake And Lung Cancer
"[He's got] plenty of courage, plenty of character and plenty of determination, and he played really well".
But Broad did reduce the West Indies from 46 for none to 53 for two.
It was not until the stroke of tea that West Indies lost their third wicket when Brathwaite edged off-spinner Moeen Ali to Stokes at slip.
A mixture of calm defiance combined with judicious attacking saw them frustrate the England attack 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 petuantly, were left increasingly disgruntled.More news: Taylor Swift's New Album 'Reputation' Coming Nov. 10
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. 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.