Cook, who'd scored a hundred in the drawn series opener at Lord's and 60 in the first innings of this match, was 48 not out.
However, a miscued hook off fast bowler Fidel Edwards, when Cook was on 42, went straight through the hands of Corey Collymore, running in at long leg.
West Indies' ground fielding had been sloppy all morning and, particularly in their position, they needed to hold every chance offered.
Kevin 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 23 not out.
West Indies' morning was summed up when Jerome Taylor let a Pietersen drive go straight through his legs at mid-off for four.
In all England, 1-0 up in the four-match series, scored 102 runs in Saturday's morning session for the loss of captain Michael Vaughan, caught and bowled by Test debutant Darren Sammy for 40.
They resumed on 34 for one, a lead of 175, with Cook 12 not out after Andrew Strauss saw his poor run continue when he'd been lbw to Edwards for nought. Vaughan was unbeaten on 10.
Sammy, the first St Lucian to play Test cricket, struck on Saturday when holding a low-right handed catch off a Vaughan drive as he followed through.
Vaughan had put on 98 with Cook and England were 99 for two, a lead of 240.
Edwards, recalled after missing the first two Tests, challenged Pietersen with some sharp deliveries before Cook edged a drive short of slip on 41.
England had a first innings lead of 141 after bowling out West Indies for 229 in reply to their 370.
Spinner Monty Panesar took four for 50, including his 50th Test wicket, and pace bowler Ryan Sidebottom three for 48.
In all, West Indies lost their last six wickets for 13 runs in 44 balls.
Brief Scores (at lunch, D-3):
England 370 and 136 for 2 (A Cook 48 n o, Vaughan 40, Pietersen 23 n o; Edwards 1 for 46, Sammy 1 for 35) Vs West Indies Ist innings 229.