Record-chaser James Anderson, Chris Jordan, Stuart Broad and James Tredwell all took wickets for England but they were then held up by Jermaine Blackwood and a typically obdurate Shivanarine Chanderpaul.
And England were denied a fifth wicket when Ben Stokes had Blackwood caught off a no-ball in the final session.
However, West Indies are still trailing by 244, having bowled England out for 399 in the morning.
The tourists resumed on 341-5 but lost four wickets for as many runs at one point, before a last-wicket stand of 38 between Jordan and Anderson took them to 399.
Stokes resumed on 71 not out but added just eight to his tally before steering a Jerome Taylor (3-90) delivery straight to Jason Holder in the gully.
Night-watchman Tredwell (8) also fell with the score on 357, caught at slip by Bravo off Holder (2-69).
England wicketkeeper Jos Buttler fell four runs later after an unusually subdued stay at the crease.
Having failed to score off any of his first 21 balls, he then edged Kemar Roach (4-94) through to keeper Denesh Ramdin.
Broad also went for a duck, caught by Blackwood at point off Roach, but Anderson made a breezy 20 before he was caught at cover off Marlon Samuels, Jordan finishing unbeaten on 21.
West Indies openers Kraigg Brathwaite and Devon Smith survived a testing three overs before lunch and took the score to 19 after the interval.
However, Anderson (1-24) then produced a trademark away swinging delivery to catch the edge of the left-handed Smith???s bat, Buttler jubilantly moving to his left to pouch the nick.
And Anderson, who led England onto the field in his 100th Test, is now just two Test wickets short of Sir Ian Botham???s England record.
Darren Bravo came in to join Brathwaite and he helped take the score to 42 before he too was caught by Buttler.
Having initially looked set to play at a delivery from Jordan (1-29), Bravo tried to take his bat away at the last moment but left it too late and feathered a catch through to the keeper.
But that was England???s last success of the afternoon session as the solid Brathwaite was joined by Samuels, who had a pair of narrow escapes.
Anderson missed a chance to run out the West Indies man when he went for a very risky single and Tredwell put down a very difficult one-handed return catch from just his third ball of the day.
Samuels??? luck ran out just after tea when he got a thin edge to a good delivery from Broad (1-46) and fell for 33, Buttler taking his third catch of the innings.
And it was 99-4 when Brathwaite???s battling knock came to an end, the opener brilliantly caught one-handed at slip by a diving Jordan after edging Tredwell (1-22) and falling for a 102-ball 39.
That brought Blackwood to the crease and he got off the mark second ball with a six over long-off before settling down to play a gritty innings.
At the other end was the 40-year-old Chanderpaul , who edged his first ball through the slips for four off Broad and then began to show off his usual solid defence and fine judgment of which balls he needed to play at and those he could safely leave.
With the score 133-4, Blackwood fended at Stokes and Cook took a good catch, high at slip, only for replays to show that the England all-rounder had overstepped the mark.
A frustrated Stokes (0-32) went on to bowl a lively spell but he was kept at bay and Chanderpaul pulled Broad away for four to bring up the 150 and the 50-run partnership.
The hugely experienced left-hander finished the day on 29 not out from 98 balls, while Blackwood has made 30 not out from 85, having finished the day with a beautiful cover-driven four in the penultimate over.