The duo put on a 246 run partnership for the fourth wicket after their side was reduced to 35 for 3. The West Indies were 329 for five at stumps, 71 runs ahead of England’s 258.
Opening batsman Brathwaite scored 134 runs, while Hope scored his maiden Test hundred and was unbeaten on 147.
The West Indies’ performance was even more commendable given they had lost the first Test match inside three days by an innings and 209 runs.
The visitors, who are down 1-0 in the series, had given a good account of themselves as a bowling unit when they dismissed the English side in the first innings for 258 runs.
The West Indies resumed on 19 for one. However, England pace bowler James Anderson soon dismissed both night watchman Devendra Bishoo and Kyle Hope cheaply to reduce the West Indies to 35 for 3.
Brathwaite, who resumed on 13 not out, successfully reviewed lbw decisions on 35 and 46. He reached his half century with a six off Moeen Ali. He went to his sixth hundred in 39 Tests with a six off the bowling of Tom Westley.
Hope survived a difficult chance on 72 before completing his first hundred in 12 Tests.
“It’s a monkey off my back,” Hope said.
“It was more about getting the team over the line, trying to bat as deep as possible in the game. You know it’s going to swing, it’s England. The ball swung a lot when I first came in to bat,” he said.