Put in to bat, England got off to a terrible start before Joe Root's composed 54 helped the 2010 champions post a competitive 155 for nine wickets.
The total seemed enough for them to successfully defend until Carlos Brathwaite hit Ben Stokes for four straight sixes to power them to a four-wicket win with two balls to spare.
"First and foremost I can't fault anything we did with the ball, on the field today, I thought we were exceptional," Morgan told reporters.
"I thought we let ourselves down tremendously with the bat. We were probably 40 short, 180-190 wicket. I thought both sides let themselves down with the bat.
"At no stage did the West Indies get ahead of where we liked them to be. We controlled them really well."
Morgan found it difficult to express how he felt after Brathwaite snatched victory from the jaws of defeat.
"I'm not sure what I am feeling at the moment. I am not quite sure it actually happened," a morose Morgan said.
"A range of different emotions. After batting and getting a score on the board and having been in control, everything felt like it ran pretty smoothly."
After their opening Super 10 defeat by West Indies, England themselves pulled off an improbable win against pre-tournament favourites South Africa, chasing down a 230-run target at Mumbai's Wankhede Stadium.
Morgan said he could now understand how the Proteas must have felt after the defeat.
The England captain believes his side have made tremendous progress in limited-overs cricket in the last 12 months since being unceremoniously dumped out of the 50-over World Cup in Australia and New Zealand.
"I am very proud of the guys regardless of what happened today," the 29-year-old Morgan said.
"I thought today was about letting us go out play with the freedom that we trusted ourselves in the whole tournament.
"The exciting thing is that I believe this is going to be the start of something special."
(Editing by Ken Ferris)
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