He shared stands of 83 for the third wicket with Kevin Pietersen and then 90 for the sixth with Matt Prior.
Talha took two wickets in four balls to put England in trouble in the seventh over after they were asked to bat first in this three-day match.
The swing bowler, included in Pakistan's Test squad here, had Andrew Strauss caught-behind with a very good delivery - and then number three Trott offered no shot to go lbw to one that nipped back.
Pietersen needed 10 balls to get off the mark, as he had in the first innings of England's previous tour match on this same ground.
But he was soon driving confidently, while Cook's outlet against pace was mostly on the back foot - before both batsmen manoeuvred comfortable runs up to lunch off spinners Raza Hasan and Yasir.
The afternoon began badly for England when Pietersen fell in the first over, picking the wrong ball to force away off the back foot and bowled by Yasir.
Bell was lbw to his second ball, attempting to sweep Yasir, and Morgan disappointingly picked out deep backward-square with a more expansive version of the same shot soon afterwards.
Yasir, in particular, was a test for all on a used pitch which was offering plenty of help for spin - albeit slowly - but Cook was equal to the task.
Prior, who missed England's first match with a bruised left ring finger, was in the mood to make the most of match batting practice before next week too - and dug in to keep Cook company and keep the tempo handy for the remainder of the session.
Prior's was the first of four wickets to fall after tea, nibbling an edge behind to a diving Sarfraz Ahmed.
Graeme Swann also took an attacking approach, and succeeded with a straight six off Yasir and two fours until he edged another attempted big shot behind off the leg-spinner.
Graham Onions, in his first match for his country for more than two years following a career-threatening back injury, fell to a one-handed catch at mid-on - and when Cook finally went caught-behind trying to cut, Strauss decided on a declaration which spared Chris Tremlett from having to bat so soon after recovering from an eye infection.
He did not have long to bowl either, before England turned to spin at both ends in conducive conditions - and in the final over of the day, Swann almost had Nasir Jamshed caught-behind on eight.