The tourists leveraged an imposing first-innings lead of 107 when they bowled England out for 219 on the third afternoon, before losing regular wickets of their own as they failed to bat their opponents out of the game.
At stumps they were 244 ahead on 137 for eight, still strongly poised but suddenly vulnerable to a fourth-innings chase.
Not deemed fit enough to play a full role with the ball due to a nagging quad injury, the all-rounder stepped up late in the day to add the sheen to a strong fightback.
Pope quickly turned his 46 not out into a a half-century but the first hour was little more than a trial for the batsmen. A paltry nine runs came off the bat in that time – outscored by 10 extras in the same period – as the three-pronged pace attack of Shaheen Afridi, Mohammad Abbas and Naseem Shah worked up a head of steam.
Buttler edged the third ball of Naseem's spell just short of second slip but after that it was a litany of plays and misses as the bowlers weaved away tirelessly.
Somehow the duo reached drinks intact and, when both hit the cover boundary after the break, there was a suggestion of better things to come.
Instead, Naseem simply upped his game again to see off Pope for 62, with a length ball which kicked and flew to gully. Remarkably that was the solitary success of the morning, Buttler and Woakes steering the score to 159 for five as scoring chances belatedly opened.
Buttler, still burdened by three missed chances behind the stumps, had scrapped hard for his 38 but fell to his second ball after lunch. Yasir Shah was his downfall, sneaking one between bat and pad to clip the top of off stump
Yasir made short work of Dom Bess, ousting him with turn and bounce before taking his fourth of the innings when Woakes heaved towards leg and hit fresh air.
England were in trouble at 170 for eight but had Broad to thank for two tail-end stands which added 49 to the total before Shadab Khan intervened on both. He finished unbeaten on 29, with three boundaries in a row off Shaheen and a slog-swept six off Yasir under his belt.
There were nine overs before tea for England to strike back and Broad did so almost instantly, Shan Masood strangled down leg for a duck for a deflating follow-up to his epic 156. Buttler, who let him off twice in the first innings, was the most relieved man on the park when the ball settled cleanly in his gloves.
James Anderson created a second chance when Abid Ali edged an away-swinger but Stokes went across Joe Root and fumbled. It was not a costly miss, with the opener reaching 20 before hacking Bess to deep midwicket.
Woakes held the catch safely but had an even greater job to do with the ball. He was the fifth bowler used but made a major impact, rousing England with the two big middle-order wickets.
First was Babar Azam for five, feeling outside off stump and giving Stokes the chance to make up for his error, and next came skipper Azhar Ali, pinned lbw full on middle stump.
Once again, the Warwickshire man was making a fair case as the team's most dangerous weapon in home conditions. Bess and Anderson failed to keep a lid on the scoring, going at almost four an over, meaning their captain had to look elsewhere for inspiration.
Dom Sibley obliged when he rushed in from point to run out Shafiq after a poor call from Mohammad Rizwan, and Stokes put concerns over his quad complaint aside to volunteer for his first bowl of the match.
He struck with his eighth attempt, forcing Rizwan back and earning a tight lbw, and when Broad picked up Shadab via DRS the tail was well and truly open.
Stokes bounced out Shaheen in the final over of the day, leaving everything to play for.