Miraz finished with a six-wicket haul on debut as England posted 293 at the Zahur Ahmed Chowdhury Stadium in Chittagong on Friday.
Miraz grabbed his sixth when Stuart Broad (13) was caught behind on review. The England pacer had been adjudged lbw earlier but he was reinstated after a review showed the ball would have missed the stumps.
The tourists began the day with an overnight score of 258-7. Debutant spinner Mehedi Hasan Miraz dazzled on the opening day.
He returned 5-64 to become the youngest Bangladesh bowler to take five wickets on debut and the third spinner in history to pick up five wickets on his first day in Test cricket.