After Bangladesh were bowled out for just 64 earlier in the day, Australia appeared well placed to build a substantial advantage.
But Shoriful produced a sensational spell with the new ball and later returned to dismantle the Australian lower order, leaving the hosts at 165/8 at stumps after 40 overs.
The left-arm quick finished with 6/35 from 11 overs, repeatedly breaking through Australia's batting line-up. His victims included Travis Head, Marnus Labuschagne, Steven Smith, Beau Webster, Pat Cummins and Mitchell Starc.
Australia's innings began positively before Shoriful struck for the first time, dismissing returning opener Travis Head for 22.
Taskin Ahmed then removed Matthew Renshaw for 8 before Shoriful struck again to send back Labuschagne for just 4.
The biggest breakthrough came when Shoriful trapped Steve Smith lbw for 42, ending a 77-run fourth-wicket partnership with Cameron Green and preventing Australia from completely taking control.
The left-armer then ran through the lower order. He dismissed Beau Webster for a duck, had captain Pat Cummins caught behind for 0, and then bowled Mitchell Starc for 1 to complete his six-for.
At the other end, Cameron Green provided Australia with their main resistance. The allrounder remained unbeaten on 51 from 81 balls, striking eight fours, and carried Australia through the final stages alongside Nathan Lyon, who was unbeaten on 10.
Bangladesh had earlier endured a disastrous start after being asked to bat first. Mitchell Starc ripped through the top order, claiming 6/12 from 10 overs, as the visitors collapsed to 64 all out in only 34 overs.
Bangladesh lost their first six wickets for just 21 runs, with Shadman Islam, Tanzid Hasan, Najmul Hossain Shanto, Mominul Haque, Litton Das and Mushfiqur Rahim all falling cheaply.
Shoriful top-scored with 14 before becoming one of Starc’s victims.
Despite the collapse, Shoriful's remarkable response with the ball has given Bangladesh something to take into the second day. Australia remain in front, but with only two wickets left and the lead standing at 101, the visitors will hope to finish the innings quickly and keep the deficit within reach.
For Shoriful, it was a dream comeback after missing the first Test through injury. Having returned to the XI in place of Ebadot Hossain, he immediately justified his selection with a spell that transformed the complexion of the match.
