Pakistan have only ever lost one Test in which they have led on first innings by at least 179 runs, against Australia at Sydney in 2009/10 when they established a 206-run advantage at that stage only to suffer a 36-run defeat.
But at least Buttler, who finished on 66 not out, and Bess (55 not out) justified their selections and gave their team a bit of confidence ahead of a potentially even more hard five-test series against top-ranked India.
Pakistan play the second Test against England at Headingly.
The pair grafted away and picked off the loose balls before fast bowler Mark Wood eventually ended a stand worth 75 runs when, having tried to "rough up" the batsmen from around the wicket, he slipped in a fuller-length delivery that Haris (39) edged to wicket-keeper Jonny Bairstow.
Pakistan again showed England how to play in their own conditions in the first Test at Lord's on Friday before suffering a blow when Babar Azam was ruled out of the remainder of this two-Test series with a wrist injury. Cook, who was playing his record-equalling 153rd consecutive Test match, scored a gutsy 70.
It was teenage leg-spinner Shadab Khan who struck next, however, when a ball that turned towards Stoneman kept low and hit the off-stump to bowl the left-handed opener for nine.
Ben Stokes unfurled a few trademark attacking strokes in his 38 but he was trapped lbw by Abbas following a review and England's last five wickets crumbled for the addition of 16 runs straight after tea.
Pakistan's Mohammad Amir celebrates after snagging the wicket of England's David Malan.
The ICC noticed that two Pakistan cricketers were wearing smartwatches during the first day of the Lord's Test and consequently announced the verdict to ban the use of the device.
England were now 91 for four and badly needed a batsman to support captain Joe Root, who had made a 93-ball fifty.
Pakistan, 350 for eight overnight, saw Amir go on to make 24 not out - with a lofted on-drive off Stuart Broad worthy of a top-order batsman one of the tailender´s four boundaries.
A frustration of the talented Root's career has been an inability to turn Test fifties into hundreds and Saturday was a case in point.
A four by Buttler, selected on the basis of his form in the Twenty20 Indian Premier League, saw him to an 89-ball fifty, with the 20-year-old Bess - in his first match at Lord's - receiving huge cheers when he reached the landmark in 93 balls.