Cook's 10-hour knock led the hosts to a first-innings total of 514 for eight declared and James Anderson dismissed Kraigg Brathwaite for nought before the tourists battled to 44 for one in reply.
Heavy rain curtailed the final session and West Indies will resume on Saturday 470 runs behind.
Resuming on 348 for three in the first day-night test in England, Cook and Dawid Malan smoothly extended their fourth-wicket partnership to 162.
Cook reached his fourth test double century with a thick edge to the third man boundary off Kemar Roach, the former England captain's 30th four.
Malan, who recorded his first test 50 in his third match, was out on the stroke of lunch for 65, edging spinner Roston Chase to Jermaine Blackwood at slip.
Ben Stokes (10), Jonny Bairstow (18) and Moeen Ali (nought) fell in quick succession as England chased runs and Cook's marathon vigil ended when he was trapped lbw by Chase, prompting the declaration.
A callow West Indies batting lineup was always likely to struggle against England's experienced pace bowling attack and Anderson had Brathwaite caught by wicketkeeper Bairstow for an eight-ball duck.
Powell, dropped by Stokes at gully off Stuart Broad, and Hope battled through to tea without further loss and they extended their partnership in the first match of the three-test series before heavy rain ended the day's play.
(Reporting by Ed Osmond,; Editing by Ken Ferris)
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