The captain and former captain eased England's concerns over how they would deal with the challenge of batting against the pink ball with a 248-run partnership for the third wicket.
Root made 136 and Cook was not out on 153 at stumps yesterday, which were called just after 9.30 pm (local time) at a flood-lit Edgbaston.
Their stand overcame some trying early times in the first day-night test in England, and first involving England, which was 39-2 in the eighth over when the pair came together.
On his way to his 13th test century, Root also became only the sixth man to score a half-century in 11 consecutive tests. AB de Villiers holds the outright record with half- centuries in 12 straight tests.
A sweep for four off a leg-side delivery from spinner Roston Chase took Root past 100 and saw him start his second series as captain as he did his first series, with a hundred.
Cook's century was his 31st, his first since losing the captaincy to Root, and first since November last year. It was his 10th score of 150 or more, equalling an England record.
The current and former skipper embraced when Root got his century, and Root then celebrated for his teammate, arms waving in the air, as he ran through for the single that took Cook to three figures.
England faced a testing few overs when the game got underway in the afternoon, with opener Mark Stoneman -- Cook's 12th opening partner since Andrew Strauss' retirement in 2012 -- out for 8 on his debut. No. 3 Tom Westley was also dismissed for 8 in his third test.
Although England still has problems to solve in its top order, finding another effective opener and a solid No. 3, and the newcomers aren't cutting it yet, there are no doubts over the lasting value of Cook and Root.
By the lunch break, which came at 4 o'clock in the afternoon, they had taken England to 108-2 and the recovery was well on its way. By tea, or rather supper time, England was 215-2.
West Indies has more day-night test experience than England having played one against Pakistan in Dubai last year. But Roach sent the second ball of this test so wide of the stumps that it went to second slip, and that set the tone for the West Indian bowling performance rather than his good deliveries to beat Stoneman and Root.
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