All-rounder Woakes was not out on 68 while opener Roy made 52 to help England chase down a 226-run target and claim the second one-day international and the three-match series having won the opener by 45 runs at the same venue on Friday.
"Jason set the tone, and got us miles ahead of the rate," said man of the match Root.
"I just told Chris to take his time to get in, and build together. He played exceptionally well, and it's a sign of a good team that we don't rely on one or two players."
But then the specialist spinners got the West Indies back into the match, triggering a slide which saw four England wickets tumbling for 16 runs.
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Ashley Nurse returned his best-ever figures of three for 34 while Devendra Bishoo claimed two for 43.
However with fast bowler Shannon Gabriel unavailable after an opening three-over spell because of a side strain, captain Jason Holder resorted to part-time spinners on a helpful track without success.
Woakes, the more adventurous of the seventh-wicket pairing in a stand that put on an unbroken 102 runs, was missed at long-on by substitute fielder Rovman Powell off Carlos Brathwaite in what was the West Indies' last real chance of staying in the game and squaring the series.
Root struck just three boundaries in a controlled innings, his only blemish being an edge off Bishoo between the wicketkeeper and slip.
"It was just about staying calm because we were so far ahead of the required rate," added Root. "Chris (Woakes) took a lot of the pressure off me with his aggression."
Fresh from claiming career-best ODI figures (4 for 40) in the first match, seamer Liam Plunkett played a key role in stifling West Indies' quest to accelerate on a bright morning.
Leg-spinner Adil Rashid, apart from showing a safe pair of hands with the catches to dispose of both Mohammed and Carter, also accounted for dangerous big-hitters Holder, via a steepling catch of his own bowling, and Carlos Brathwaite.
Fast bowler Steven Finn did the early damage to the West Indies innings on the same pitch used for the series-opener.
Poor shot selection contributed to Evin Lewis' demise, a leading edge giving a simple catch to Sam Billings at cover, while a miscued pull by new batsman Kieran Powell presented Finn with the catch off his own bowling that took him to the milestone of 100 ODI wickets.
"We need to seize the opportunities that come our way," Holder noted in reflecting on batsmen failing to capitalise on good starts and also chances missed in the field in both matches in Antigua.
"We keep losing too many wickets to poor shots at the top.