Usman Khawaja was unbeaten on 50 and Travis Head 34 not out at close with Australia still needing an improbable 326 runs for victory while Pakistan need seven wickets on the final day for a 1-0 lead in the two-match series.
Medium pacer Mohammad Abbas took all three wickets for 26 runs. Pakistan had declared their second innings on 181-6 after resuming at 45-3.
Jon Holland took 3-83 while Nathan Lyon finished with 2-58.
Earlier Pakistan declared their second innings for 181 for six and set a target of 462 for Australia to chase on day four of the first Test in Dubai on Wednesday.
Pakistan lost two early wickets but still built a formidable lead over Australia. They went to the interval at 155 for five in their second innings, with Asad Shafiq on 22 and Babar Azam unbeaten on 23.
Pakistan made 482 in their first innings while in reply Australia were dismissed for 202.
Jon Holland, who took the key wicket of Azhar Ali for four on Tuesday, dismissed opener Imam-ul-Haq for 48, caught and bowled off a flighted delivery.
Debutant Marnus Labuschagne then had Haris Sohail caught close to the slips for 39, leaving Pakistan at 110-4. Sohail and Imam had added 65 runs for the fourth wicket.
Holland has figures of 3-71 from 16 overs while off-spinner Nathan Lyon has yet to take a wicket.
In the first session, Holland and Labuschagne dismissed Imam and Haris as Pakistan continued to add to their overnight score of 45.
Despite being jolted by spinner Holland’s two key wickets of Mohammad Hafeez (17) and Azhar Ali (four), Pakistan closed day three on 45-3, having an overall lead of 325 with seven second-innings wickets intact to force a win.
Opener Imam was unbeaten on 23 while nightwatchman Asif was dismissed for nought, unable to guard Azhar from coming to bat in a crucial stage.
Test debutant Bilal Asif snatched six wickets to spark a spectacular Australian collapse from 142-0 to 202 all out and put Pakistan in firm command of the first Test in Dubai on Tuesday.
The 33-year-old off-spinner overshadowed his more experienced team-mate Yasir Shah by finishing with 6-36 as his efforts, the third best figures for a Pakistani bowler on Test debut, derailed Australia on a pitch which started to assist spinners on the third day.
It was a day highlighted by Asif who triggered a memorable Australian collapse against spin in Asia to mark a long-awaited Test debut.
“I knew that my chance would come, so I waited and waited and thanks God I have done myself proud by getting six wickets on debut, it´s a tremendous feeling,” said Asif.
“I was struggling a bit in the first session, but all my team-mates and coaches gave me confidence and that increased my energy which helped me get six wickets.”
Asif, who had previously played three one-day internationals in 2015, came into his own after pacer Mohammad Abbas removed Finch following an opening stand of 142.
In a spell of 22 balls Asif took four wickets as Australia slumped to 180-5 by tea.
The second session belonged to Asif, who dismissed Khawaja, Shaun Marsh (seven), Travis Head (nought) and Marnus Labuschagne (nought) in the space of 29 runs.
Asif´s feats on debut had only been bettered by fast bowler Mohammad Zahid (7-66 vs New Zealand in Rawalpindi in 1994) and spinner Mohammad Nazir (7-99 against New Zealand in Karachi in 1969).
Asif is the 11th Pakistani bowler to take five or more wickets in an innings on Test debut.
Before Asif’s brilliance, Abbas had forced Finch to drive straight into the hands of Asad Shafiq at short mid-on.
Khawaja hit eight fours before miscuing a premeditated sweep and was caught at short leg while Finch had five boundaries and a six in his knock.
Leg-spinner Yasir was unlucky not to be amongst wickets, having dropped Mitchell Marsh off his own bowling for two and wicketkeeper Sarfraz Ahmed missing a stump in the first session with Khawaja on 17.
Yasir went wicketless after 28 probing overs.
Earlier in the day, Finch was more confident as he punched two boundaries and a six off Yasir before sweeping Asif for a boundary to reach his half-century.
Khawaja, the Pakistan-born batsman, improved upon his previous best score in Asia of 26, made against Sri Lanka at Galle in 2011, by reaching his 13th Test half-century.
The second and final Test starts in Abu Dhabi from October 16. Both teams will also play three T20 internationals after Tests.