Skipper Shakib Al Hasan (25 not out) and nightwatchman Shafiul Islam, unbeaten on nought, were at the crease at the close with Bangladesh 95 runs ahead and only four wickets in hand.
Earlier, all-rounder Tim Bresnan compiled a Test-best 91 to lead England to a first-innings total of 496.
Bangladesh opener Tamim Iqbal then blazed his way to a half-century before being caught by Stuart Broad at point off spinner Graeme Swann for 52.
Tamim had his fair share of luck, having been dropped three times with spinner James Tredwell the unlucky bowler on each occasion.
Wicketkeeper Matt Prior put the opener down when he was on 15 before Tredwell failed to hold a sharp return catch on 26.
Jonathan Trott also dropped the simplest of chances at point with Tamim on 47.
Tredwell was, though, lucky to remove Zunaed Siddique (34) when his drive hit the toe of captain Alastair Cook at silly-point and looped up for a simple catch for the bowler.
Bresnan had Mahmudullah caught by Prior for six before Swann dismissed debutant Jahirul Islam.
Jahirul, who made a duck in the first innings, opened up with two sixes and eventually made 43.
Broad, who dismissed opener Imrul Kayes for four, returned later to clean bowl vice-captain Mushfiqur Rahim for three.
Earlier, England resumed on 440 for eight and batted nearly 20 overs to add 56 runs.
Left-arm spinner Abdur Razzak (3-132) claimed both tail-end wickets to fall, both after stumpings by wicketkeeper Rahim, while Shakib finished with figures of four for 124.
Bresnan fell nine runs short of his maiden test century after hitting six fours in his 261-ball knock.
