Earlier, hosts were all out for 220 in their first innings of the final match of the series after losing their last nine wickets for just 49 runs, with off-spinner Moeen Ali taking 5-57.
Opener Tamim Iqbal made a quickfire century, scoring 104 off 147 balls, while Mominul Haque scored 66 as the hosts surged to what should have been a commanding score of 171 for one.
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But Tamim's dismissal, lbw to Ali, sparked a spectacular collapse, with seven of Bangladesh's batsmen failing to make double figures at Dhaka's Sher-e-Bangla National Stadium.
The all-rounders Ben Stokes and Chris Woakes shared the other five wickets as the tourists tightened the noose on a Bangladeshi side still reeling from the disappointment of losing the first Test by just 22 runs.
Tamim and Haque completely dominated the morning session after Woakes earned England an early breakthrough removing opener Imrul Kayes for one, with Ben Duckett taking a spectacular catch at point.
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Tamim smashed 12 boundaries, but he became Ali's first victim of the day shortly after driving the same bowler for back-to-back fours through cover to bring up his third century in six Tests against England.
It ended his 170-run stand with Haque, who had also looked solid until he was bowled by Ali for 66.
Bangladesh skipper Mushfiqur Rahim had called on his team before the start of the Test to show more consistency and learn how to drive home their advantage at key points in matches.
But the rapid succession of wickets in the afternoon session pointed to Bangladesh's continued frailty in high-pressure situations, severely damaging their hopes of squaring the two-match series.
England made two changes to the team that won the first Test last Monday in Chittagong, with left-arm spinner Zafar Ansari handed a Test debut for Gareth Batty and paceman Steven Finn coming in for Stuart Broad who has been rested.
Bangladesh picked all-rounder Shuvagata Hom ahead of Shafiul Islam in their only change. They are playing with just one specialist fast bowler.
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