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.
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 equalling the England record shared by Leonard Hutton, Walter Hammond and Kevin Pietersen.
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.
On the field, left-hander Cook continued to look as if he was having a net against an outclassed attack who continued to feed his areas of strength.
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.
His pull off Miguel Cummins, one of four seamers in the attack, saw Malan complete a 112 ball-fifty.
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.
Kemar Roach, the luckless bowler, deserved better support having previously gone past the outside edge of both batsmen.
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.
On Thursday, Cook and captain Joe Root both scored hundreds as England dominated the first day of the Test.
Together with Root (136), Cook put on 248 for the third wicket.
The past and present England skippers had come together with their side faltering at 39 for two after both debutant Mark Stoneman and Tom Westley had been dismissed for eight.
Root’s 11th fifty in successive Tests was a new England record and just one shy of South Africa star A.B. de Villiers’s mark of 12.
For Cook, already England’s leading Test run-scorer and century-maker, this was his 31st hundred at this level.
ENGLAND (1st Innings, overnight 348-3):
A. Cook not out 213
M. Stoneman b Roach 8
T. Westley lbw b Cummins 8
J. Root b Roach 136
D. Malan c Blackwood b Chase 65
EXTRAS (LB-10, W-3, NB-6) 19
TOTAL (for four wkts, 117.2 overs) 449
TO BAT: B. Stokes, J. Bairstow, M. Ali, T. Roland-Jones, S. Broad, J. Anderson
FALL OF WKTS: 1-14, 2-39, 3-287, 4-449.
BOWLING: Roach 28-8-86-2 (3w); Joseph 22-3-109-0 (2nb); Cummins 23-3-86-1 (3nb); Holder 21.3-4-79-0 (1nb); Chase 16.5-1-73-1; Brathwaite 6-0-6-0.