Chasing a target of 273, England seemed to be cruising when they reached 100-0 at tea on the third day with Cook unbeaten on 39 alongside Ben Duckett, 56 not out.
Young Bangladeshi off-spinner Mehedi Hasan struck off the first ball after the break to remove Duckett, ending their partnership that triggered a dramatic collapse.
England soon crumbled to 164 all out in less than two hours, suffering their first-ever defeat to Bangladesh in 10 Tests and sharing the two-Test series 1-1.
Cook downplayed the suggestion it was his toughest career defeat, but said sloppy fielding made things difficult for the side.
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England had the chance to limit their target when their bowlers created umpteen chances in the morning session.
But they dropped catches and missed a crucial stumping to allow Bangladesh to build on their overnight 152-3 and eventually reach 296 runs in their second innings.
"I've had some tough ones," Cook said, referring to other career losses.
"I thought 270 would have been a real good chase to be honest. They got 30 too many today.
"But in these conditions on spinning wickets, their bowlers are good."
Mehedi and Shakib Al Hasan and Taijul Islam claimed all 20 England wickets in the Test, forcing Cook to admit they made the difference in the series.
"I think they did out-bowl our spinners. But, you know, that's credit to those guys.
"We're not hiding behind the fact that we haven't got world-class spinners. We've got guys who can bowl some really good balls.
"You've got to reverse it or skid it or spin the ball. That's the challenge of Test cricket," said Cook.