The hosts now lead by 52 runs after a near-flawless batting display.
Opener Mahmudul Hasan Joy was the star of the day, remaining unbeaten on 169, while Mominul Haque provided strong support with 80 not out.
Earlier, fellow opener Shadman Islam departed after scoring 80, but not before adding 168 runs in a solid opening stand with Joy.
Ireland, who resumed the morning on 270 for 8, could add only 16 runs before being bowled out for 286.
The Tiger openers then made full use of the favorable conditions, laying a perfect platform for the middle order.
With a commanding position and wickets in hand, Bangladesh will look to extend their lead further and tighten their grip on the Test on Day 3.
