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Yasir made a quickfire 33 off just 24 balls, including five fours and a six.
A brief session saw Pakistan add 32 runs in 16 balls before they were dismissed for 169 in their second innings.
Only twice has a team chased more than 200 to win in the fourth innings of a Test at Old Trafford, with England making 294-4 against New Zealand in 2008 and 231-3 against the West Indies at the Manchester ground in 2004.
Yasir had already taken 4-66 in his primary role as a leg-spinner during England's modest first-innings 219.
Pakistan resumed on 137-8, a lead of 244, after they had lost four wickets late on Saturday, with star all-rounder Ben Stokes striking twice shortly before stumps to help drag England back into the match.
But Stokes did not bowl on Saturday, with England captain Joe Root opting for a combination of Jofra Archer and Stuart Broad.
Yasir, 12 not out overnight, extended the lead past 250 by taking 10 runs off the first three balls of Saturday's play from fast bowler Archer.
A clipped legside two was followed by a powerful pull for four well in front of square from a short ball that did not get up.
And when Archer dropped short outside off stump, Yasir cleverly uppercut him high over the slips for another boundary, with 11 runs in total coming off the first over.
Yasir then promptly clubbed Broad's first ball of the day through midwicket for four.
And he surpassed that stroke when, two balls later, he hoisted Broad for a magnificent six.
Broad had Yasir caught behind but by then the damage had been done.
Last man Naseem Shah then played a shot worthy of a top order batsman when he got off the mark first ball by lofting Broad straight back over his head for four before he was bowled by Archer.
England are trying to end a run of losing the first Test in each of their last five series.
This is the first of a three-Test contest, with Pakistan having not lost a series against England since 2010.
Pakistan 1st Innings 326 (Shan Masood 156, Babar Azam 69; S Broad 3-54, J Archer 3-59)
England 1st Innings 219 (O Pope 62; Yasir Shah 4-66)
Pakistan 2nd Innings (overnight: 137-8)
Shan Masood c Buttler b Broad 0
Abid Ali c Woakes b Bess 20
Azhar Ali lbw b Woakes 18
Babar Azam c Stokes b Woakes 5
Asad Shafiq run out (Sibley) 29
Mohammad Rizwan lbw b Stokes 27
Shadab Khan lbw b Broad 15
Yasir Shah c Buttler b Broad 33
Shaheen Afridi c Burns b Stokes 2
Mohammad Abbas not out 3
Naseem Shah b Archer 4
Extras (b4, lb5, nb4) 13
Total (all out, 46.4 overs, 223 mins) 169
Fall of wickets: 1-6 (Masood), 2-33 (Abid), 3-48 (Babar), 4-63 (Azhar), 5-101 (Shafiq), 6-120 (Rizwan), 7-122 (Shadab), 8-137 (Afridi), 9-158 (Yasir), 10-169 (Naseem)
Bowling: Anderson 9-2-34-0 (1nb); Broad 10-3-37-3 (1nb); Archer 6.4-0-27-1 (1nb); Bess 12-2-40-1; Woakes 5-1-11-2; Stokes 4-1-11-2 (1nb);
England: Rory Burns, Dom Sibley, Joe Root (capt), Ben Stokes, Ollie Pope, Jos Buttler (wkt), Chris Woakes, Dom Bess, Jofra Archer, Stuart Broad, James Anderson
Match position: England set 277 to win
Toss: Pakistan
Umpires: Richard Kettleborough (ENG), Richard Illingworth (ENG)
TV umpire: Michael Gough (ENG)
Match referee: Chris Broad (ENG)
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