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.