Teenager Mehedi Hasan starred with a five-for on his Bangladesh debut as England battled to 258-7 at stumps on day one of the first Test in Chittagong.
Moeen Ali (68) led England's recovery from a poor start, remarkably prevailing in five reviews - two rejected attempts by the Tigers to have him dismissed and three successful challenges of his own - to make an invaluable half century.
And Jonny Bairstow, closing in on Andy Flower's record for Test runs made by a wicketkeeper in a calendar year, added 52.
But both of England's top scorers were among the five players who fell to the unexpected star of the show, as 18-year-old Mehedi (5-64) shone brightly on his Test debut.
Ben Duckett's first spell in Test cricket lasted 35 balls, the newcomer making 14 runs, including two boundaries, before the grip and turn of his fellow debutant found the middle and off stumps.
Cook followed an over later, the captain playing onto his stumps attempting a sweep off Shakib Al Hasan (2-46) and departing for four.
It was an underwhelming contribution for the captain, particularly given he had rushed to the subcontinent following the birth of his second child, his brief stay at the crease enough to set a new record for his country of 134 Test appearances, surpassing Alec Stewart.
Bangladesh claimed their third scalp in the space of three overs when Gary Ballance fell lbw to Mehedi for just one run following a successful review from Mushfiqur Rahim.
Mehedi claimed his biggest victim yet just after lunch, drawing an edge from Joe Root (40) that was taken by Sabbir Rahman at slip.
Ben Stokes (18) got a start but was too slow to prevent a late-dipping Shakib delivery from rearing up to remove the top of off stump.
Moeen made the most of his reprieves to steady the ship alongside Bairstow, before Mehedi struck again, forcing the all-rounder to play at a ball that drifted in from outside off and eliciting a nick that was snaffled by Mushfiqur.
And Mehedi sealed his memorable debut five-for with a delivery that skidded on to beat Bairstow's defence and clatter into off stump.
England reached the close of play with Chris Woakes (36 not out) and Adil Rashid (5 no) at the crease, having rallied to earn a respectable position going into day two.