For the third time in the last seven matches, a Test in Australia is on track to finish inside two days after Starc and captain Pat Cummins (3-10) ripped through Bangladesh, who batted poorly.
Starc took 6-12 as Bangladesh were dismissed for just 64 on day one.
On Sunday he was just as lethal and his 4-39 took his match tally to 10 wickets for a fourth time in his career.
Bangladesh were 8-95 at tea, still needing 51 runs to make Australia bat again.
After Australia took a 146-run lead, left-arm paceman Starc, in his second over, ripped out opener Shadman Islam (one) and Mominul Haque with successive deliveries to be on a hat-trick for the second time in the match.
After his early strikes, Starc was in the game again when Tanzid Hasan (nine) skied a Cummins delivery straight to him at fine leg.
Captain Najmul Hossain Shanto (31) gifted Nathan Lyon his first wicket of the Test with Starc gobbling up the catch at long off.
Starc then snapped up Litton Das for his second duck of the match and the back of the Bangladesh batting had been broken.
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 an unplayable inswinger that hit him on the toe in front, a delivery that Pakistan great Wasim Akram would have been proud of.
Green and Lyon resumed on day two with Australia 8-165 and the pair added a valuable 38 for the ninth wicket, with Lyon making a cameo of 32.
Bangladesh's batting in this Test has been awful, but they confirmed 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.
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