Along with fellow Bajan Kraigg Brathwaite's fine batting - he added 95 to his first-innings century - and the evident reawakened spirit in the team, this was an emphatic answer to Windies greats like paceman Curtly Ambrose who had called their Edgbaston performance "pathetic".
"We're test cricketers for a reason".
Jermaine Blackwood then muscled the visitors towards the finishing line with Hope, smashing a typically thunderous 41 off 45 balls including two sixes.
Hope, who had scored his maiden test hundred with 147 in the first innings, batted for 211 balls for his unbeaten 118, playing with remarkable skill and composure to guide the Windies to victory.
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.
Stuart Law was a proud Windies coach after his side rallied from their Edgbaston nightmare to beat England in the second Test at Headingley.
When Ali came to the crease, England had just lost three wickets in six overs to offspinner Roston Chase and was 327-7 and just 158 runs ahead.
Openers Brathwaite and Kieran Powell came through unscathed, reaching 5-0, but Root's decision - one which former England spinner Graeme Swann said stamped the Yorkshireman's authority on his side - was notable in its boldness compared to predecessors Alastair Cook and Andrew Strauss.
But the West Indies will head into the final session still in with a chance of just their fourth away Test win, excluding fixtures against Bangladesh and Zimbabwe, in 88 matches spanning 20 years.
"I always believed in this group", Holder told Sky Sports. I'm extremely proud of these two guys, they worked very hard and it's paying off.
"After the Birmingham Test match where everybody was saying we were the worst team that ever arrived on these shores, we've scrapped with England, we've battled them and we've taken it into a fifth day". He's been going through a form dip but, hopefully, the pretty 20s are gone and the pretty 120s are here to stay.
"We felt like their heads went down pretty quickly after tea", left-handed batsman Ali told reporters.
It proved the one that got away as Brathwaite rode his luck to contribute a further match-shaping 91 runs in a crucial stand with Hope, who was also grassed by Cook after he had passed his hundred, the final nail in England's coffin.
"[He's got] plenty of courage, plenty of character and plenty of determination - for a 24-year-old, that's a lot to have".