At stumps, England, looking to level the four-Test series at 1-1, were 98 for one in their second innings.
That gave them a lead of 489 runs, with no side having made more in the fourth innings to win a Test than the West Indies' 418 for seven against Australia at St John's in May 2003.
Cook, who made a first innings hundred, was 49 not out and vice-captain Joe Root, whose 254 in the same innings was his Test-best score, unbeaten on 23.
Shortly before the close, England opener Alex Hales (24) was caught behind off Mohammad Amir.
Pakistan were dismissed for just 198 in reply to England's 589 for eight declared -- a huge deficit of 391 runs.
England 98-1 against Pakistan, lead by 489 at 3rd day close
Yet Cook, on a day where four rain interruptions caused 27 overs to be lost, decided to bat again.
That reduced the time England had to bowl Pakistan out again, although there were, in fairness, still more than two days remaining in the match.
"It was a pretty simple decision to bat again," England assistant coach Paul Farbrace told Sky Sports.
"We thought we'd bat while the wicket was still good, rather than put ourselves under pressure when it gets worse.
"It could have gone either way, but I think it's the right decision and hopefully we'll be proven right."
Pakistan head coach Mickey Arthur said the tourists simply had to bat better in their second innings -- whenever that came.
"I was disappointed with our first innings," Arthur told reporters.
"We need more discipline outside off stump. We were nicking off that 'fifth stump' line a lot, which was particularly disappointing," he added.
Tall order: Well-Rooted England bat Pakistan into a corner
Asked if not being made to follow-on had made it more likely that Pakistan could draw the game, Arthur replied: "It probably gives us a bit better chance but that's not an inflammatory headline, as irrespective of any declaration we have to bat damn well."
Pakistan's first innings saw England pace-bowling all-rounder Chris Woakes take four for 67, while Root held four catches at second slip.
Misbahul Haq, the Pakistan captain, provided a lesson in application to the rest of the top order with an innings-high 52 and shared a ninth-wicket stand of 60 with Wahab Riaz, who made a Test-best 39.
Pakistan, who resumed on 57 for four, saw their collapse continue before lunch on Sunday as they lost further four wickets for 62 runs.
But with Cook opting against using experienced paceman James Anderson and Stuart Broad after lunch, Pakistan added 79 runs in the second session.
Pakistan started Sunday's play with opener Shan Masood (30 not out) and Misbah (one not out).
Cook proud to lead from front as Root runs riot
Anderson, England's all-time leading wicket-taker, marked his return to Test cricket on his Lancashire home ground, after missing Pakistan's series-opening 75-run win at Lord's last week with a shoulder injury, by taking the first wicket to fall on Sunday when Masood (39) was caught by Root.
Pakistan were now 71 for five, their total at the rain break.
Shortly after play resumed, Asad Shafiq (four) drove loosely at a Broad slower ball and lobbed a catch to Hales at point.
Sarfraz Ahmed came 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.
Woakes, who took 11 wickets at Lord's and three more on Saturday, struck again when Sarfraz, playing a rare defensive shot, nicked Ben Stokes to Root.
Misbah, who made a fine hundred at Lord's, completed a 108-ball fifty before, sweeping at off-spinner Moeen 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 also featuring five fours.
But left-arm quick Wahab was hit on the forearm by Woakes when batting and did not bowl later on Sunday.
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