England were 348-6 in their second innings at lunch on the fourth day of the second Test, a lead of 307 runs, with Brook dismissed for 109 and Root 81 not out.
The Yorkshire duo shared a partnership of 189 in 41 overs after coming together Saturday when England were just 99 runs ahead at 140-3 after the loss in quick succession of Ollie Pope (51) and Ben Duckett (76).
But England lost three wickets for 19 runs shortly before the end of Sunday's first session, with skipper Ben Stokes (eight) and Jamie Smith (six) both falling.
England started the day's play on 248-3, already 207 runs in front.
Brook was 71 not out and Root unbeaten on 37, with England bidding to take an unassailable 2-0 lead in a three-match series after a dominant innings and 114 run-win in the first Test at Lord's.
Conditions remained good for batting on Sunday, with Brook soon adding to his tally when he cover-drove Jayden Seales' fifth ball of the day for a four that sped across a quick outfield.
Brook's straight-driven four off Seales saw him into the 90s before two more boundaries off another paceman in Alzarri Joseph -- an on-drive and a square-drive behind point -- took him to 99.
The 25-year-old's quick single off Alzarri Joseph ensured Brook got to three figures, with his fifth hundred in 14 Tests but first on home soil coming in just 118 balls, including 12 fours.
It was at Trent Bridge where England launched their 'Bazball' era two years ago by chasing down a target of 299 against New Zealand in 50 overs.
But star batsman Root showed there was still a place for traditional red-ball batting, with the 33-year-old former England captain completing a 91-ball fifty, including four fours.
Brook's impressive innings finally came to an end when he was caught behind off a flat-footed drive from a lifting Seales ball that moved off the seam.
Stokes then carelessly hooked Seales straight to Alzarri Joseph at long leg. But at 341-5, England were exactly 300 runs ahead.
West Indies, however, struck again on the stroke of lunch when off-spinner Kevin Sinclair had Smith edging to opposing wicketkeeper Joshua Da Silva.
