However, the tourists would have been in an even stronger position had they not dropped England captain Joe Root when the star batsman had made just eight on his Yorkshire home ground.
Root, who won the toss, was 24 not out at lunch, with Dawid Malan unbeaten on three.
The West Indies recalled fast bowler Gabriel and leg- spinner Davendra Bishoo after an innings and 209-run defeat inside three days in the inaugural day/night Test in England at Edgbaston last week that left them 1-0 down in this three- match series.
Gabriel had missed the Edgbaston match after bowling 24 no-balls in a preceding 'pink-ball' warm-up fixture against Derbyshire.
But back in standard hours and with the red ball at their disposal, he and the West Indies attack responded well to captain Jason Holder's calls for greater bowling discipline after England had piled up 514 for eight declared at Edgbaston, with Cook making 243 and Root 136.
It was a different story early on at Headingley when left-hander Cook, England's all-time leading Test run-scorer, out for 11 after edging a Gabriel delivery from just back of a length low to Kyle Hope at third slip.
But they were each dismissed cheaply Friday.
England's 19 for one became 26 for two when Westley, an Essex team-mate of Cook, was plumb lbw to Kemar Roach for three after being struck on the back leg.
Stoneman, Cook's 12th England opening partner in five years since the retirement of Andrew Strauss and playing just his second Test, then exited for 19, caught behind off an inside edge against Roach, who had lunch figures of two for 21 in nine overs.
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