West Indies were 13 for one in reply to England's imposing 514 for eight declared, a deficit of 501 runs, at tea on the second day.
Cook's 243 was the cornerstone of England's total and the former captain's dismissal prompted current skipper Joe Root to declare.
That left the West Indies with a tricky nine overs to bat in twilight before the interval.
That was time enough for opener Kraigg Brathwaite to be caught behind for a duck off Anderson, England's all-time leading Test wicket-taker.
Powell was five not out at tea, with Test debutant Kyle Hope, in with West Indies nought for one wicket, unbeaten on seven.
England resumed in an immensely strong position of 348 for three in the first of this three-match series and the 50th Test at Edgbaston.
Cook was 153 not out -- his tenth score of 150 or more in Tests.
Together with Root, who made 136 after winning the toss, he'd put on 248 for the third wicket.
On the field, left-hander Cook -- England's all-time leading Test run-scorer and century-maker -- continued to look as if he was having a 'net' against an outclassed attack.
Malan, who in marked contrast to Cook is appearing in just his third Test, provided steady support during a century stand that took the total beyond 400.
His pull off Miguel Cummins, one of four seamers in the attack, saw Malan complete a 112 ball-fifty.
But West Indies did have a success on the stroke of lunch when off-spinner Roston Chase had left-hander Malan caught at slip by Jermaine Blackwood to end a partnership of 162.
Cook was 213 not out at lunch. But his chances of surpassing his highest Test score of 294, made against India at Edgbaston six years ago, were effectively undone with a rush of wickets at the other end.
Moeen Ali, England's man of the series in their recent 3 -1 win over South Africa then carelessly holed out off Chase for a duck.
That left England 506 for seven.
Cook, who now had tailender Toby Roland-Jones for company, fell to Chase, who took four for 113, when he missed an attempted legside flick.
For the second time this innings, the West Indies had to review a not out decision from South Africa umpire Marais Erasmus to confirm the dismissal.
But with replays indicating the ball would have hit leg stump, Cook's marathon effort of more than nine hours and his first Test century in 17 innings was finally at an end.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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