At tea, England were 262 for four in their second innings -- a lead of 159 runs -- after losing both Root (62) and James Vince (42).
Root's Yorkshire team-mates Gary Ballance (21 not out) and Jonny Bairstow (two not out) were unbeaten after a session where England managed just 79 runs in 26 overs. With the four-match series all square at 1-1, England resumed on 120 without loss, 17 runs in front.
But Pakistan removed both batsmen inside Saturday's first five overs during a dramatic burst of two wickets for no runs in nine balls.
Left-hander Cook (66) pushed out to Sohail Khan and a diving Shah held an excellent catch at point. Mohammad Amir then turned England's 126 for one into 126 for two when Hales (54) edged a seaming delivery from the left-arm quick and Younis Khan held a difficult low chance going to his right at second slip.
The latter shot has often proved Vince's undoing in his brief Test career. But it is also one of the Hampshire batsman's best strokes and it enabled him to get off the mark with a stylish four off Amir.
Root, however, had a reprieve on 25 when he edged Rahat Ali only for Mohammad Hafeez to drop the low, two-handed, chance at first slip. It was tough on left-arm quick Rahat, who reeled off five successive maidens in a probing spell of seven overs costing just seven runs.
Vince followed up with a fine leg glance for four off Shah. Sohail, altering his line, almost got Root to play on.
But it was Shah, bowling into the rough outside leg stump, who made the breakthrough.
Root, not for the first time this season, mistimed a sweep and gave a simple catch to Hafeez at short fine leg.
Root's exit ended a stand of 95 in 36 overs with Vince that had taken England to 221 for three. Vince, yet to make a fifty in nine Test innings, had been composed in equalling his highest score at this level of 42.
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