Cook batted through the day for his unbeaten 153, sharing 248 with skipper Root, who made 136, as England racked up 348-3 at Edgbaston.
Opener Mark Stoneman managed eight on his debut and Tom Westley fell for the same score, but Dawid Malan is 27 not out.
Kemar Roach took two wickets for West Indies, who were poor with the ball and in the field.
Much had been made about how the pink ball would behave, but the tourists did not bowl with enough accuracy to exploit any movement that might have been on offer.
