The defending champions were skittled for just 55 by an excellent England attack, with Adil Rashid taking 4-2 and Mills 2-17, after Moeen Ali had done the early damage with a spell of 2-17 from four overs.
England stuttered a little in their innings at 39-4, but Morgan (seven*) and Jos Buttler (24*) saw them safely home with 70 balls to spare.
Mills, 29, has returned to the England fold after a lengthy absence and Morgan told Sky Sports he was pleased to have the left-arm fast bowler back at his disposal.
“I’m delighted for ‘Big T’,” said Morgan. “He’s had an incredibly unfortunate journey throughout his career right from the start, but now he’s as good as I’ve seen him, both fit and bowling fit as well.
“His fielding is also drastically improved, which is a huge contribution, so I’m absolutely delighted that not only is he back on the field for us, but contributing in the manner that we’ve seen him over the past in his Sussex shirt.”
Morgan was obviously buoyant at England’s opening match win.
“I think it’s as good as it gets,” said Morgan. “To start a world tournament like that, or a campaign like that, I think full credit has to go to our bowling unit.
“I don’t think the wicket was all that flush to bat on but still even the start West Indies got, we made inroads, they came back at us and we stuck to our guns. The guys were very disciplined.
“I thought we fielded well, we took all of our chances and built pressure.”
England play Bangladesh in their second game tomorrow (Wednesday).