Bell made 138, his 10th Test century, and was well supported by Tim Bresnan, who hit an unbeaten 74 as the pair shared a sixth-wicket stand of 143 before England eased past Bangladesh's first-innings 419.
Bresnan was fortunate not to be given out on five when Imrul Kayes caught what replays suggested was an inside edge off the bowling of Shakib Al Hasan.
Bell then survived an lbw decision off the bowling of Abdur Razzak when on 82. Bangladesh missed another chance to dismiss Bell on 119, with Kayes dropping him at square leg off paceman Shafiul Islam.
Bell was finally out going for a slog sweep against captain Shakib, only managing to sky the ball to midwicket where debutant Jahirul Islam completed the catch.
"It was tough out there. We knew coming into this morning that we had to play well because Bangladesh so far have played some good cricket," Bell said.
"It was nice to make a contribution when it was needed. I hope Bressy can go on tomorrow to score a hundred - it was nice to get us into a position where hopefully we can get a few more runs, put some pressure on Bangladesh and hopefully push for the win."
Shakib then ran out Graeme Swann for six before Mahmudullah trapped Stuart Broad lbw for three as the hosts finished the day with some kind of success.
"I think both teams are now under some kind of pressure," said spinner Shakib after claiming four for 99.
"If we take their last two wickets within 30 runs they will get only a 50-run lead. Then we have to bat very well which is pressure for us and whatever runs we score they have to chase it which will be a pressure for them on the fifth day." Bangladesh tasted early success on the third morning, Shakib bowling Jonathan Trott with the England opener failing to add to his overnight 64.
Matt Prior joined Bell at the crease and survived a close lbw appeal off Rubel Hossain while on nine but he regained focus and started padding the run rate after the Bangladeshi spinners had kept England on the back foot for the first hour.
Prior completed his 50 with a boundary over mid-on off Mahmudullah and smashed Shakib for two fours in the next over off successive balls before the left-arm spinner had his revenge.
Shakib bowled Prior for 62, ending his 98-run partnership with Bell.
Bangladesh trail 1-0 in the two-Test series.
