Pakistan were dismissed for just 198 in reply to England's huge 589 for eight declared -- a deficit of 391 runs.
Yet Cook decided against asking Pakistan to bat again.
England had the fifth biggest first-innings lead without enforcing the follow-on in Test history.
Two of those instances had come in 'timeless' Tests, while the others were when Australia led England by 445 runs in Brisbane in 2006 and 398 at Adelaide in 2013 -- with the Australians winning both of those Ashes matches.
Another rain interruption forced an early tea, with England 11 without loss in their second innings.
Cook, who made 105 in the first innings, was nine not out and Alex Hales two not out.
Earlier, England all-rounder Chris Woakes took four for 67, while Joe Root followed his first-innings 254, his Test-best score, with four catches at second slip.
Misbah-ul-Haq, the Pakistan captain, top-scored for his side with 52 and shared a ninth-wicket stand of 60 with Wahab Riaz, who made a Test-best 39.
But with Cook opting against using the experienced new ball duo of James Anderson and Stuart Broad after lunch, Pakistan added 79 runs in the second session.
- Wickets tumble -
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Pakistan started Sunday's play with opener Shan Masood 30 not out and Misbah one not out.
Left-handed opener Masood had held firm for more than two hours in all.
But having moved on to 39, Masood exited in familiar fashion when he edged Anderson to Root.
Shortly after play resumed, Asad Shafiq fell for four when he drove loosely at a Broad slower ball and lobbed a catch to Hales at point.
Sarfraz Ahmed may have come in at 76 for six but the wicket-keeper stuck to his attacking game and got off the mark when he cut Broad over Hales's head for four.
By contrast it took Misbah 47 balls to score the first boundary of his innings, a square-drive off Anderson.
Woakes, who took 11 wickets in defeat at Lord's, saw his first over Sunday cost 11 runs as Sarfraz hit two well-struck boundaries.
Misbah completed a 108-ball fifty before, sweeping once too often at off-spinner Ali, he was caught at short fine leg by Cook.
Wahab hoisted Ali for six. But going for another big hit off the spinner, he holed out to end a 61-ball innings that also featured five fours.
