After starting day three at 150 for three, England had the chance to dismiss Bangladesh cheaply but they dropped at least four catches and allowed the home team to post 296.
Cook and Joe Root handed reprieves to Imrul Kayes while Moeen Ali and Ben Duckett were the other culprits as England were set a target of 273 runs in the final innings.
However, they capitulated to the spin combination of Mehedi Hasan and Shakib Al Hasan and were dismissed for 164, allowing Bangladesh to register their maiden Test victory against England.
Cook also rued the fact that England lacked a world-class spinner and is hoping for his bowlers to learn from the two tough games they had in Bangladesh.
"We're not blessed at the moment with a world-class spinner," said Cook.
"We've got to make do with the best we've got.
"Our spinners have to try to hold a length better. Hopefully that can happen with a lot of hard work.
"I thought Zafar Ansari improved as the game went on. He was a little bit nervous, understandably so for the first game of his Test career, but he got better and bowled really well today.
"He could have taken six-for quite easily if we had taken our chances. Today, we should have been chasing 230. Those half chances need to stick in low scoring games.
"Adil Rashid bowled well at the end to take four wickets and Moeen Ali was good in the first innings.
"There are glimpses of the right stuff, though there is improvement to be made."
At one stage during the chase, England were coasting with a score for 100 for naught with Duckett scoring a blistering half-century and was well supported by Cook. However, once Duckett was dismissed, the floodgates opened and England lost their remaining nine wickets in a span of 64 runs.
Cook blamed the batting collapse to the inexperience of the players in the sub continent conditions.
"Today we showed our inexperience in these conditions," said Cook. "A lot of these guys have not played many Test matches over here and when that ball got rolling we found it very hard to stop.
"I was out there for a lot of it as well and I couldn't stop that ball. I've played a bit of cricket in the sub-continent, and I've found these two games particularly tough.
"The first 20, 30 balls are so difficult to start with - there's pressure, men round the bat, the crowd get into it, and it's a very unique pressure.
"Plus you've got one ball skidding on and then the next one ragging past your stumps. It will have opened a lot of the guys' eyes to that."
The Essex batsman, however, lauded Duckett for his innings which had seven boundaries and a six.
"The top order have struggled, we can't hide from that. But what was really pleasing today though, was the way Duckett played.
"It has been incredibly tough for him. I've been around the block a bit, but he said to me, 'I've never opened the batting before against two spinners'.
"Suddenly you have to do it on big turning pitches, and it's hard work - today you saw the way he'd have loved to have played all series. He probably worked out his method a little bit more, and it was really good to see."
England now will head to India where they will play five-match Test series which will be followed by three One-Day Internationals and as many Twenty20 Internationals.
© Cricket World 2016