England, 1-0 up in the four-match series, were 228 for three in their second innings at tea on the third day of the third Test, a lead of 369.
Cook, who'd scored a hundred in the drawn series opener at Lord's and 60 in the first innings of this match, was 89 not out after sharing a third-wicket stand of 122 with Kevin Pietersen, who made 68.
Pietersen, who followed up his century at Lord's with a Test-best 226 in England's innings and 283 run win at Headingley - West Indies' heaviest Test innings defeat - was out hit wicket in unlucky fashion.
Trying to hook a short ball from Dwayne Bravo, Pietersen's helmet fell off, despite being attached to a chin strap, and hit the stumps.
West Indies, whose ground fielding was poor throughout, missed Cook on 42 and Pietersen on 51.
England resumed after lunch on 136 for two with Cook 48 not out and Pietersen unbeaten on 23.
Left-hander Cook then completed a 96-ball fifty with five fours.
But by then the 22-year-old Essex batsman was fortunate to still be at the crease after a miscued hook off Fidel Edwards went straight through the hands of Corey Collymore, running in at long leg.
Occasional leg-spinner Shivnarine Chanderpaul posed problems and Pietersen, on 32, edged him onto Chris Gayle's boot at first slip.
Gayle, who came into this match with a side injury that threatened his ability to bowl off-spin, was brought on in the 51st over.
South Africa-born Pietersen, 26, took a single off Gayle to complete a 79-ball fifty with seven fours.
But soon afterwards he chipped Gayle to mid-wicket where Runako Morton, running back, dropped the two-handed catch.
West Indies poor fielding was exemplified when Jerome Taylor let a Pietersen drive go straight through his legs at mid-off for four.
The only wicket to fall before lunch saw England captain Michael Vaughan caught and bowled by Test debutant Darren Sammy for 40.
England had a first innings lead of 141 after dismissing West Indies for 229 in reply to their 370, the tourists losing their last six wickets for 13 runs.
Brief Scores: (at tea, D-3)
England Ist innings 370 and 228 for 3 (Cook 89 n o, Pietersen 68, Vaughan 40; Bravo 1 for 14, Sammy 1 for 35, Edwards 1 for 46) Vs West Indies Ist Innings 229.