Fellow opener Rob Yates also cracked 191 in an extraordinary first-wicket partnership of 343 with outstanding skipper Davies, whose unbeaten 226 came from 274 balls with 25 fours and three sixes.
To compound the misery for promoted Durham, who had won the toss, Will Rhodes reached 60 not out to take the hosts past maximum batting points and to 490 for one before the close.
Champions Surrey took command against Somerset at The Oval despite a century from Tom Lammonby.
Lammonby (100) and Matt Renshaw (87) got Somerset off to a fine start by putting on 178 for the second wicket but the loss of seven wickets for 20 runs undermined their efforts.
Captain Lewis Gregory (50) lifted the score to 285 but Cameron Steel (four for 50) wrapped up the innings before Surrey openers Rory Burns and Dom Sibley chiselled 42 off the lead without loss.
Joe Clarke continued his excellent start to the season with another century as Nottinghamshire closed on 305 for six against Worcestershire at Trent Bridge.
Following up his ton against Essex last week, Clarke hit 105 and shared in a third-wicket stand of 133 with Ben Slater (70) as the hosts recovered from the loss of two early wickets.
Dean Elgar (120) scored his first Essex century and Matt Critchley (103no) also reached three figures against Kent at Chelmsford.
Jordan Cox added 67 and Michael Pepper 49 as the hosts amassed 421 for six at stumps.
Hampshire were 305 for six at the end of day one against Lancashire at the Utilita Bowl.
Liam Dawson was unbeaten on 61 after solid efforts from Tom Prest (85), James Vince (56) and Nick Gubbins (50).
Australia’s Nathan Lyon (two for 97) bowled 32 overs as Lancashire kept themselves in the game.
In Division Two, spinner Alex Thomson took a career-best seven for 65 as Derbyshire bowled out Glamorgan for 287 in Cardiff. Kiran Carlson was the hosts’ top-scorer with 74.
Derbyshire were 46 for one in response.
Emilio Gay’s unbeaten 165 helped Northamptonshire to 311 for three against Middlesex at Wantage Road. Captain Luke Procter added 73 in a second-wicket partnership of 189.
Pakistan captain Shan Masood’s 140 was the mainstay of Yorkshire’s 326 against Gloucestershire at Bristol. Jonathan Tattersall (58) and Matt Milnes (51) also contributed but Zaman Akhter’s five for 89 helped the hosts hit back before they ended the day on 28 for two.
Liam Trevaskis was 82 not out as Leicestershire closed on 326 for eight against Sussex at Grace Road. Rishi Patel (87) and Peter Handscomb (51) also hit half-centuries.
Hurraira was expected to get a chance when the two-match series gets under way in Rawalpindi on Wednesday, having impressed on the domestic stage for Islamabad United.
He scored 218 against a Bangladesh High Performance XI in a four-day match last month, also receiving his first international call-up for a tour of Sri Lanka.
He was not selected for his red-ball debut during that series, however, and he has also been overlooked for this week's first meeting with Bangladesh, despite Imam-ul-Haq being rested.
"We have plenty of riches in our opening batting," Masood said ahead of the Test. "Imam-ul-Haq isn't here for this series.
"We don't limit ourselves to 17 players. We have 20 to 25 players who are part of our squad.
"We initially felt that it was our ideal chance to check Muhammad Hurraira and see what he could offer the Pakistan team after doing so well in domestic cricket in the last three or four years.
"We rested Imam in this series. Saim, who is in good touch now, showed potential in the second innings against Australia.
"As a team, you have to send a message of continuity. So we will try to back the players. We want to give him a fair chance."
Masood also said the expected conditions at the Rawalpindi Cricket Stadium had influenced the decision to select Mohammad Ali to lead the bowling attack.
"You have to look at a few factors," Masood said. "You have to take into account Pakistan's position in the WTC.
"We at least have to win the home matches. Secondly, you have to look at the ground conditions. Thirdly, we have to look at our team combination.
"I think Mohammad Ali gave us the surprise element, we have been seeing it for a long time."
Miraz, who took five wickets during the opening Test, again starred with the ball for Bangladesh, crucially dismissing Saim Ayub (58) and Shan Masood (57).
Taskin Ahmed (3-57), who was brought in for the injured Shoriful Islam, got Bangladesh off to a flying start with the wicket of Abdullah Shafique (0) in the very first over.
Ayub and Masood steadied the ship to have the hosts 99-1 at lunch, only for the latter to fall soon after, which resulted in the wickets beginning to tumble.
Babar Azam again struggled with the bat, scoring 31 from 77 deliveries, with Khurram Shahzad (12), Mohammad Ali (two) and Abrar Ahmed (nine) all falling victim to Miraz's impressive bowling display.
Bangladesh openers Shadman Islam and Zakir Hasan ended the day at the crease with the visitors 10 without loss as they look to confirm the series triumph on Sunday.
Data Debrief: Miraz the man
Miraz's impressive bowling display proved to be the difference on the second day, putting Bangladesh in a strong position in the series.
It was also the 10th five-wicket haul of his career, three of which have come away from home.
Pakistan has unveiled a spin-dominant squad for the upcoming two-match home Test series against West Indies, aiming to exploit the visitors’ well-documented struggles against spin bowling.
