Awais was unbeaten on 85 while Fazal got to 37 not out as Pakistan went to the close of play on day two on 179-1 in the first innings.
The 21-year-old Awais was involved a 106-run opening partnership before Imam-ul-Haq fell lbw to the spin of Mehidy Hasan Miraz for 45.
That was the only breakthrough for the home side though as 23-year-old Fazal shared in an unbroken stand of 73 to finish off a strong day for the tourists.
Bangladesh had resumed the second morning on 301-4 but lost six wickets for just 112 runs as the Pakistan pace-attack, led by Mohammad Abbas, hit back.
Abbas finished with figures of 5-92 in 34 overs while left-arm quick Shaheen Shah Afridi took 3-113 as the visitors limited the damage after Bangladesh enjoyed a good opening day in the two-match series.
It could have much better for Pakistan after overnight batter Mushfiqur Rahim fell for 71 but Taskin Ahmed, batting at number 10, played a quick-fire cameo of 28 off 19 balls to take the hosts over the 400-run mark.
Bangladesh captain Najmul Hossain Shanto said fierce competition within the side's bowling unit had become the driving force behind their historic 2-0 Test series sweep over Pakistan after a 78-run victory in Sylhet on Wednesday.
Cricket Australia have made several bold calls in the announcement of their touring squad for six ODIs and three T20Is in Pakistan and Bangladesh.
Captain Najmul Hasan Shanto scored a century as Bangladesh edged Pakistan with the bat on the first day of the two-match Test series in Mirpur.
Australia are set to be missing several key personnel when they tour Pakistan for three ODIs starting later this month.
