After Australia took a 146-run lead, the left-arm paceman, in his second over, had opener Shadman Islam (one) caught in the gully by Marnus Labuschagne and then clean bowled Mominul Haque with his next delivery.
Starc on day one on Saturday had captured 6-12 as Bangladesh were dismissed for just 64.
After his early strikes on day two, Starc was in the game again when Tanzid (nine) skied a Pat Cummins delivery straight to him at fine leg.
Bangladesh went to lunch at 3-39 with captain Najmul Hossain Shanto (23no) their best hope of posting a first-innings lead.
Earlier, Bangladesh pace bowler Shoriful Islam returned all-time best away figures for his country.
Allrounder Cameron Green was ninth man out for 67 to take his series aggregate to 184, the best of either side, and certainly secure his spot on the plane to South Africa for the three-Test series in October.
Left-arm quick Shoriful finished with 7-48 after returning his career-best figures in an innings before Australia were dismissed for 210.
They were also the best innings figures away for a Bangladesh bowler, surpassing left-arm spinner Taijul Islam's 7-138 in Harare in June.
The delivery that dismissed Green was a snorting inswinger that hit him on the toe and deflected to third man.
On review the ball was clearly crashing into middle stump, an unplayable delivery that Pakistan great Wasim Akram would have been proud of.
Green and Nathan Lyon resumed on Sunday with Australia on 8-165 and the pair added a valuable 38 for the ninth wicket with Lyon making a cameo of 32.
The big question at lunch was whether the sold-out crowd at Great Barrier Reef Arena would get to see a full day's play, considering 18 wickets tumbled on the opening day.
They certainly got a glimpse of a future star in Shoriful, who missed the first Test with a side strain.
The 190cm tall tearaway is just 25 and appears set to lead the attack for years to come. He has looked up to Starc and certainly bowled with the same kind of threat.
"When I was playing age-level cricket I saw him and I liked him," Shoriful said of Starc after taking six wickets on the first day.
"I am playing with him, so I'm excited. He got six and I also got six, so (I am) happy."
The quietly-spoken giant would be happier still after finishing with seven wickets.
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