England were 449 for four at lunch on the second day, with opening batsman and former captain Cook 213 not out after more than eight hours at the crease.
It was his second double century at this level at Edgbaston, with Cook having made 294 -- his highest Test score -- against India at the Birmingham ground six years ago.
The one disappointment for England was that they lost Dawid Malan for 65 to what became the last ball of the session following the Middlesex batsman's maiden Test fifty.
Cook was 153 not out -- his tenth score of 150 or more in Tests equalling the England record shared by Leonard Hutton, Walter Hammond and Kevin Pietersen.
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Together with current skipper Joe Root, who made 136 after winning the toss, he'd put on 248 for the third wicket.
There was a sombre start to Friday's proceedings with players, officials and spectators observing a minute's silence observed in memory of the victims of Thursday's terror attacks in Spain.
It was not long before the 32-year-old Essex batsman glanced West Indies captain Jason Holder down to fine leg for four and clipped young paceman Alzarri Joseph off his pads during another sunny daytime session where conditions were ideal for batting.
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.
It was a sound effort given he'd come in under lights and was batting against the new pink ball but equally any judgement on whether Malan was doing enough to cement his place in England's squad for their upcoming Ashes defence in Australia had to take into account the modest quality of the bowling he was facing.
Cook then went to his 200, his fourth Test double century, when a misfield by Kyle Hope on the third man rope -- an error symptomatic of a generally sloppy team display in the field -- allowed the ball to trickle through for the opener's 30th four in 339 balls.
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.