Brandon King hit an unbeaten 52 in the chase as victory was wrapped up in comprehensive fashion with 17 balls to spare.
There are four more T20 matches to come between the sides, before a three-match Test series in the Caribbean gets underway at the start of March.
England would have hoped that a change of formats, and Sam Billings aside a change of personnel, would bring a much-needed boost to results, but right from the off it was a tale that fans have become all too familiar with over the winter.
Jason Roy was bowled in the first over, before Jason Holder had Tom Banton caught behind and Moeen Ali dismissed the very next ball for a golden duck as England fell to 10-3.
James Vince hit Sheldon Cottrell for a six before the bowler got his revenge immediately as a long-hop was smashed straight to short cover. Jet-setting Billings was stumped for just two, before Liam Dawson was run out as the shambolic scenes in the middle continued.
Morgan became the third batsman to hit a six and get out the next ball, as he fell for 17 off 29 balls to leave his 49-7 with eight overs remaining.
Some lusty hitting from Chris Jordan got England above their lowest ever T20 score, but he fell for a top score of 28 and Holder then took the last two wickets to finish with figures of 3.4-1-7-4.
A target of 104 never looked like being enough, as Hope and King took West Indies to 39-0 at the end of the powerplay.
Adil Rashid had Hope stumped for 20 to give England a slight opening, while Dawson got through a tidy four overs for just 12 runs.
But that was to be the only joy for the bowlers, as Nicholas Pooran finished on an unbeaten 27 and King hit the winning run through point off the first ball of the 18th over to take the lead in the series.