At lunch, England were 131 for four in their second innings and needing just 60 more runs to reach their victory target of 191 in the opening match of this three-Test series.
Left-handed opener Cook was 53 not out, his fifty coming when he late cut off-spinner Marlon Samuels for an eighth four in 78 balls.
Together with Ian Bell (34 not out), he'd so far added an unbroken 74 for the fifth wicket after England had faltered at 57 for four.
England resumed on 10 for two after Kemar Roach had taken two wickets for seven runs in eight balls to remove England captain, and first innings century-maker, Andrew Strauss, and nightwatchman James Anderson on Sunday.
Both Cook and Jonathan Trott had yet to score.
Roach struck again Monday to remove Trott with a good length ball that squared him up and took the edge with West Indies captain Darren Sammy, diving to his left, holding a good catch at second slip.
And 13 also proved an unlucky number for Trott's fellow South Africa-born batsman Kevin Pietersen.
He had just pulled Test debutant Shannon Gabriel's third delivery of the innings for four when, to the fast bowler's next ball ball, he tried to repeat the stroke and got a bottom edge to wicket-keeper Denesh Ramdin.
England, the world's number ranked Test side, were in trouble but Cook got them going again with a square-cut four off Roach.
And Bell then drove a Gabriel full-toss down the ground.
That West Indies -- who hadn't won a Test match outside the Caribbean against major opposition since defeating South Africa in Port Elizabeth in 2007 and came into this game with a record of just two wins in 30 matches -- still had a shot at victory owed much to Shivnarine Chanderpaul.
Officially the world's best batsman, the Guyana left-hander made 91 in a second innings total of 345 to follow his first innings 87 not out.
Together with Samuels (86) he put on 157 for the fifth wicket and, with Ramdin and Sammy adding runs down the order, the West Indies gave their bowlers a target to defend and a chance of a first win in 15 Tests in England.
