Extending an excellent run of form from the previous ODI series in India in January, the England captain stroked a polished 107 to pace the tourists to a competitive 296 for six.
Plunkett and Woakes then claimed four wickets each to thwart the home side's effort for an upset victory.
Half-centuries by Jason Mohammed and Jonathan Carter were not enough as the West Indies were dismissed for 251 in the 48th over, Plunkett taking the final wicket to finish with his best ODI figures of four for 40.
"It was tough at the start out there but the rewards eventually came by sticking it out," said Morgan.
"I thought it was a dominant performance overall by us and we've set a high standard for the remaining two matches."
Accelerating from a painstaking start, Morgan timed the innings to perfection in lifting his team to a competitive total after they were put in to bat on a rain-interrupted morning at North Sound.
He faced 116 deliveries and struck 11 fours and two sixes after coming to the crease with his side in early trouble after fast bowler Shannon Gabriel removed opening batsman Jason Roy and new Test skipper Joe Root in successive overs.
That dismissal brought in Ben Stokes and the aggressive all-rounder did not disappoint, belting 55 off 61 deliveries with three sixes and seizing the initiative from the West Indies with a 110-run fifth-wicket partnership, ended by a catch at long-on off leg-spinner Devendra Bishoo.
Morgan reached three figures in fine style, hoisting Carlos Brathwaite for a huge six over midwicket in the 48th over before the bowler had the last laugh, running out the left-hander in the final over of the innings.
However, England's perseverance paid off and when Mohammed was run out for a top score of 72 in the 42nd over, the die was cast for the West Indies despite a few bold hits by Carlos Brathwaite and Ashley Nurse.
"I thought we competed well throughout the match but just too many wickets to soft dismissals," said West Indies captain Jason Holder.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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