Set 326 for victory, Hope anchored the West Indies reply and then produced some savage hitting to guide his team home with seven balls remaining, finishing on 109 not out.
Hope smashed three sixes off the penultimate over off the hapless Sam Curran to reach three figures and seal victory.
It was West Indies' highest-ever ODI run chase against England, surpassing the 286 they managed at Lord's in 2004.
England were hoping to put their World Cup failure behind them and when West Indies slumped to 213-5 in the 38th over it appeared that they were in control.
But Hope, in partnership with the explosive Romario Shepherd, turned the match on its head as they put on 89 in 51 balls for the sixth wicket.
Shepherd was eventually out for 48 having hit three sixes but by then Hope was well and truly in the groove.
West Indies has earlier started well in their chase with Alick Athanazen (66) and Brandon King (35) putting on a century for the opening wicket but England hit back with Rehan Ahmed taking two wickets for 40 in 10 tight overs.
England's bowlers did a decent job at hauling West Indies back but Curran came in for some mighty punishment, conceding 98 runs -- the most ever for England in ODIs.
"I've played a lot of ODI cricket, always back myself to win in any position," Hope said. "(Shepherd) was amazing, one of those guys you can depend on. We got the win, want to continue the series on a high."
England captain Jos Buttler won the toss and opted to bat on a two-paced pitch and openers Phil Salt and Will Jacks smashed 77 for the opening wicket in 8.2 overs.
They did not fully capitalise on such a great start, however, with a succession of batsmen getting set but unable to really capitalise apart from Harry Brook who made 71.
Buttler was disappointing as he fell for three off 13 balls while Zak Crawley was run out for 48.
Some late hitting by Curran and Brydon Carse put on 66 for England's eighth wicket as they moved past the 300 mark.
The pick of the West Indies bowling was Gudakesh Motie who took two for 49 while Oshane Thomas and Shepherd also took two wickets on an encouraging day for the hosts.
For England, there were some encouraging signs but Buttler said they would have to learn some lessons.
"Thought it was fantastic batting effort. Loved the intent, and they set the tone at the top," he said.
"Brilliant game, credit to Shai and Romario, some fantastic sixes and we'll learn a lot. Thought Rehan bowled really well and there was a lot to be proud of."
The next match takes place on Wednesday.
