In response Alex Hales struck 43 as England started strongly but it fell to 68 for four and was all out for 155.
Windies skipper Carlos Brathwaite said it was a challenging game, given the conditions and praised his players for pulling off the win.
Alex Hales ensured England got off to a fine start in their reply, hitting 43 off just 17 balls as he looks to stake a claim for the Ashes, but Windies took regular wickets and showed the form which enabled them to become World T20 champions previous year.
"I can see both sides of it, of course guys want to see him play", said Morgan.
Jerome Taylor then gave West Indies the ideal start and silenced the capacity crowd, when he had Roy caught at backward point with the first ball of the chase.
The hosts needed 26 off the last over and Brathwaite, fittingly, ended the match when he bowled Liam Plunkett.
Brathwaite didn't not feature in the preceding Test series, which England claimed 2-1 after winning the 3rd Test at Lord's.
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West Indies opener Evin Lewis of Trinidad and Tobago bats during his innings of 51 off 24 balls, the highest individual score in yesterday's NatWest T20 global against England at Emirates Durham ICG in Chester-le-Street, England, yesterday.
On a cold and wet evening in Durham, West Indies were put into bat but their innings was ignited by Chris Gayle  - who struck his 100th six in the shortest format of the sport - and Evin Lewis .
But fast bowler Plunkett (three for 27) and leg-spinner Adil Rashid, whose figures of three for 25 were his best at this level, kept West Indies to under 200.
Chris Gayle smashed 4 sixes in his 21 balls knock of 40 and became the first man to 100 T20I sixes.
Follow all of the action with our ball-by-ball scorecard below.
The Trinidadian, however, was out soon afterwards when he drove Plunkett to Root at mid-off.
England and West Indies begin a five-match one-day worldwide series in Manchester on Tuesday.
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