The West Indies, having chosen to bat first, posted 179 for five wickets in their 20 overs.
Chris Gayle set the tone with a crunching 58 from 35 balls, before Johnson Charles, his opening partner, took centre stage, finishing with 84 from 56.
It was a solid total, and one that looked far out of range when England lost two wickets in the first three balls of their reply without scoring a run.
But though always behind the required run rate, they refused to let the West Indies coast to victory, and still had a chance going into the final over of the match.
But Morgan and Alex Hales (68 from 51) could not summon up any more heroics.
England are bottom of their group after one match by virtue of their net run rate, and are likely to need to beat both New Zealand Sri Lanka in their remaining two games if they are to advance to the semi-finals.
England made a single change from the side which were skittled by India in their previous match, putting spinner Samit Patel in for Tim Bresnan.
The West Indies welcomed back Dwayne Bravo, while Samuel Badree came in to the bowling line-up as they looked to test England with spin.
Batting first, Gayle set the tone by smashing three boundaries from the first over he faced. Charles took his time by comparison, facing several dot balls and struggling to get going.
It mattered little to Gayle what his partner was doing, or indeed what the bowlers were doing - when the powerplay overs were completed and the spinners on, the left-hander welcomed Patel with a trio of sixes.
England must have feared the worst when Steven Finn dropped Charles in the deep off the bowling of Graeme Swann with the score on 103 - but the very next ball, Gayle picked Finn out and the Middlesex seamer clung on.
Charles, however, stepped up into the aggressor's role, hitting with abandon and finishing with 10 fours and three sixes.
England finished strongly, with Stuart Broad leading from the front to finish with figures of two for 26 from his four overs.
The final total of 179 looked strong but not unassailable - New Zealand and Sri Lanka had earlier both scored 174 on the same wicket, only for the hosts to win the match in the Super Over.
But Ravi Rampaul struck early, with Craig Kieswetter top-edging an attempted pull to Kieron Pollard on the off side without scoring, and Luke Wright following the next ball, feathering a short ball to Gayle at slip.
At 0-2 England required a salvage operation. Hales and Jonny Bairstow began that, putting on 55 for the third wicket. But Bairstow scored slowly, struggling to keep the strike ticking over against the spinners, and departing at the midpoint of the innings with England requiring 12.5 runs an over for victory.
That they kept in the game for as long as they did owed much to Morgan, who rode his luck but also played expansively and quickly.
Hales, 35 not out when Morgan joined him, settled into the role of Morgan's foil, and played his part in a partnership worth 107 runs in just 58 balls.
The West Indies bowled just five overs of seam in the course of the match, with part-time spinners Gayle and Marlon Samuels called upon to bowl at crucial times.
Gayle's overs cost just 27 runs and yielded the wicket of Bairstow, while Samuels held his nerve in the final six balls to deny England the unlikeliest of come-from-behind victories.
The defending champions will rue their start, but have just two days to fix it, with defeat to the Black Caps potentially terminal to their hopes of reaching the last four.