In their first positive start of the series, Duckett completed his maiden Test fifty to reach 56 not out at tea on the third day while Cook remained unbeaten on 39 as the visitors came within 173 runs of a clean sweep.
Adil Rashid and Ben Stokes did their part with the ball earlier, sharing seven wickets between them to help the visitors dismiss Bangladesh for 296 runs in the second innings.
Shuvagata Home was unbeaten on 25 as Bangladesh added 28 runs with the last three wickets resuming the play at the lunch break with 268-7.
England had the chance to dismiss Bangladesh with a cheaper total had they not dropped a series of catches, which allowed Bangladesh to build on their overnight 152-3
Imrul Kayes was the first wicket to fall on Sunday as he departed for 78 but not before being dropped twice in the slips.
Moeen finally trapped Kayes' leg before but England proved unable to grasp their opportunities.
Jonny Bairstow missed a simple stumping chance off Ansari when Shakib Al Hasan was on 16.
Ansari was denied again by Duckett, who dropped a simple catch at deep midwicket to give Shakib a second lease of life on 23.
Skipper Mushfiqur Rahim also had his share of luck when he was dropped by Steven Finn off Ansari on six.
Rashid began his haul bowling Shakib for 41 and Rahim fell before another run had been added.
The 25-year-old had left the field in the evening session on the second day and it had been uncertain whether he would take any further part in the game.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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