Younis's 218 was the centrepiece of Pakistan's first-innings 542 all out.
His sixth innings of 200 or more in Test cricket was a dramatic way for the 38-year-old to end a run of low scores.
Younis had managed just 122 runs in six previous innings this series, with a best of 33 in Pakistan's first Test win at Lord's.
Leg-spinner Shah then took three wickets for four runs in 22 balls as England, in their second innings, slumped to 88 for four at the close -- still runs 126 behind.
England now needed something remarkable if they were to deny Pakistan victory with two days left in the game.
If the four-match series ends 2-2, Pakistan could go top of the world Test rankings but England must avoid defeat at The Oval to keep their hopes of replacing Australia at the summit alive.
England, 214 runs adrift on first innings, suffered a setback early in their second knock when captain Alastair Cook (seven) was squared up by a lifting Wahab Riaz delivery and debutant Iftikhar Ahmed, going to his right, held a good catch at first slip.
Saturday saw him fall lbw for 12 to Shah's sixth ball of the day, with England wasting a review in the process.
His exit left Nottinghamshire opener Hales with a series tally of just 145 runs at an average of 18.12.
James Vince, yet to make a Test fifty, was in even greater need of a score to cement his place in the side.
But instead he was out for nought when he chipped Shah, England's tormentor-in-chief across London at Lord's, to Pakistan captain Misbah-ul-Haq at short extra-cover.
Joe Root, one of England's key batsmen along with Cook, then fell for 39 when he played back to a quicker Shah delivery and was lbw.
England, perhaps because it was Root, reviewed but it was another wasted challenge and the hosts were 74 for four, with Shah's stumps figures of three for 15 in seven overs a testament to his effectiveness.
Pakistan resumed on 340 for six after Asad Shafiq (109) and Younis (101 not out) had both compiled impressive centuries.
But they lost two wickets in two balls soon afterwards to
Next ball, Bairstow chipped Riaz straight to Azhar in the covers to end a 127-ball innings featuring eight fours.
Bairstow has now scored 992 Test runs in 2016 and is in sight of breaking Zimbabwe great Andy Flower's record for most Test runs scored by a wicket-keeper in a calendar year of 1,045 set in 2000.
Shah, who took 10 wickets during Pakistan's 75-run win in the first Test across London at Lord's, enjoyed further success in the capital when Stuart Broad (five) reverse swept-him straight to Younis at first slip -- an extraordinary shot given England's position.
England then saw fast bowler Finn leave the field injured after pulling up just two balls into the second over of Pakistan's chase with what appeared to be a hamstring injury.
This was Pakistan's first Test series in England since the controversial 2010 tour, which saw then captain Salman Butt and pace bowlers Mohammad Asif and Mohammad Amir handed five-year bans and jail terms for deliberately bowling no-balls in a spot-fixing scandal during that year's Lord's Test.
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