The teenage offspinner produced big turns with accuracy to grab 5-64 but England fought back to reach 254-7 at stumps on a spin-dominated day at the Zahur Ahmed Chowdhury Stadium in Chittagong on Thursday.
Gritty half-centuries from Moeen Ali (68) and Jonny Bairstow (52) - both dismissed by Miraz - rescued the tourists, who were tottering at 106-5 at one stage
Moeen, however, led a charmed life as he was involved in five reviews on either side of the lunch break. He was adjudged lbw three times off Shakib Al Hasan but every time the batsman had the decision overturned on review.
Despite looking uncomfortable against spin, he paired up in two fifty-plus stands to resurrect England.
England sent in debutant Ben Duckett to open with Alastair Cook, who became the most-capped England Test cricketer with his 134th appearance.
His maiden Test wicket was that of Duckett who was completely beaten by the flight and turn.
Mushfiqur Rahim opted for spin from both ends and the move paid immediate dividend with Shakib dismissing Cook (4) with his second delivery.
Miraz then trapped Gary Ballance (1) in front on review as England slumped to 21-3 losing three wickets in the space of as many runs.
Moeen then joined up with Joe Root (40) and put up 62 runs for the fourth wicket to help his side recover from the early setback before the teams went for lunch with England on 81-3.
Although all the other batsmen had cold feet facing spin, Root stroked the ball around with ease and grace. He cracked five crisp boundaries in his fluent 49-ball knock before Miraz took him out.
Moeen then patiently got himself in, playing out 100 deliveries for his first 24 runs, and joined up with Bairstow to rescue their team. The pair soldiered on to make sure England lose no more wickets before tea.
But Miraz proved to be a thorn when he ended Moeen's watch inducing an edge to Mushfiqur and break the crucial 88-run partnership early in the final session.
Bairstow, who also rode his luck for 52, then added 43 more with Chris Woakes (36 not out) to build on Moeen's work. Bairstow continued a stellar year in Tests by cracking his eighth half-century.
He became the second wicketkeeper-batsman after Andy Flower (1045 runs) to reach a thousand Test runs in a calendar year.
The track, however, can also offer a lot for England, who are playing three spinners, when Bangladesh will come out to bat.
In addition to the spin of Moeen and Adil Rashid, England also included Gareth Batty in their Test lineup after 11 years.