Duckett, Pope and Brook help to extend England's lead over Windies on day three
Duckett scored 76, Pope 51 and Brook 71 not out as England fought back well after West Indies took a first-innings lead of 41 thanks to an entertaining last-wicket partnership of 71 between Joshua Da Silva and Shamar Joseph.
England lost opener Zak Crawley for three when he was unluckily run out at the non-striker's end after Duckett's drive was parried on to the stumps by the hand of bowler Jayden Seales.
Duckett struck 11 fours before he was trapped lbw by Alzarri Joseph and Pope contributed again following his first-innings century before being caught at gully by Kevin Sinclair off Joseph.
Brook and Joe Root, however, weathered the storm in an unbroken stand of 108 and Root was 37 not out at the close.
"It was a good day's cricket, it went back and forth a bit, and that partnership at the end was a crucial one for us in terms of getting a grip of the game," England seamer Chris Woakes told the BBC.
England picked up four wickets in the morning and looked set to establish a first-innings lead before Da Silva and Shamar Joseph guided West Indies to 457 all out to leave the match finely poised.
Da Silva finished unbeaten on 82, including three sixes, and Joseph hit two maximums, one smashing tiles on the pavilion roof, in his 33 before he was caught by Gus Atkinson off Mark Wood.
After West Indies had resumed on 351-5, Jason Holder was caught by wicketkeeper Jamie Smith off Woakes for 27 and Sinclair slashed Atkinson to gully where Brook held a sharp catch.
Alzarri Joseph edged Woakes to Smith for 10 and the experienced England seamer bowled Seales first ball with a fine delivery.
"It was a tough old slog but I thought we stuck at it well. I think we'd have taken getting those last five wickets for 90 this morning," Woakes said.
"You can always look back and think 'could we have done this or that' but we were pretty committed to what we wanted to do. I didn't feel like they were getting away from us."
Shamar Joseph averted the hat-trick and Da Silva moved to his fifth test fifty as the pair combined to give a fluctuating game another twist.
England won the first match of the three-test series by an innings and 114 runs at Lord's.