Dawid Malan was also caught by Sarfraz off Hasan for six to leave England struggling at 3-43.
Cook and Jonny Bairstow dug in with a partnership of 57 before Bairstow was bowled for 27 by Faheem Ashraf.
Cook, playing in a record-equalling 153rd consecutive test, was bowled by an excellent delivery from Mohammad Amir to end an innings lasting over three hours.
The experienced left-hander, who hit 14 fours, looked crestfallen after missing out on a 33rd test century and he trudged slowly back to the pavilion to a standing ovation.
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.
Jos Buttler, on 14, edged Hasan to second slip where Asad Shafiq took a sharp catch.
Dom Bess, making his debut, scored five before giving Shafiq another catch off Abbas, Stuart Broad was lbw to Abbas for nought and Mark Wood was caught by a diving Amir at mid-on for seven.
England needed early wickets and Broad snared Pakistan opener Imam-ul-Haq lbw for four, but the hosts failed to make another breakthrough under the floodlights.
Azhar Ali (18 not out) and Haris Sohail (21 not out) were largely untroubled by the England fast bowlers, although Azhar was dropped in the slips on 16 by Stokes off Wood to complete a poor day for Root’s side at the start of the two- match series.
Cook acknowledges that Pakistan are on top but believes England are not yet down and out.
“We’re behind a little bit, I think Pakistan have had the better day,” Cook told Sky Sports.
“But with a five-day Test match, you’ve got the chance to come back and it starts tomorrow morning.
“Suddenly 50 for one can very quickly become 80 for four and the game’s back even-stevens. Credit to Pakistan, they bowled well and they showed some fight with the bat.
“We were probably 60 or 70 short on that wicket and it’s probably going to be a first innings versus fourth innings game. With it drying out, chasing 250 on that last day could be hard work.
“We’ve got to get ourselves in a position to do that, that’s a long way ahead.”
Morning & Afternoon Newsletter