Left-handed opener Cook was closing in on his 29th Test century when he played round his front pad to the paceman, bowling from around the wicket, and was was plumb lbw for 85.
Cook's exit ended a 173-ball innings including nine fours.
It also halted a fifth-wicket stand of 80 with Jonny Bairstow, 44 not out at tea, that had helped England recover from 84 for four.
Moeen Ali, fresh from his Test-best 155 not out in the second Test at the Riverside, was one not out.
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England had already won this three-match series at 2-0 up.
But Cook insisted in the build-up to this match that England were determined to correct their habit of losing 'dead' Tests in series they'd already won following heavy defeats at the end of victorious campaigns at home to Australia in 2015 and away to South Africa earlier this year.
Fielding an unchanged side, England made a serene start after Cook won the toss in ideal sunny batting conditions on a placid Lord's pitch, albeit one offering a touch of seam movement.