At tea on the fourth and penultimate day, the hosts were 479 for seven in 125 overs, a lead of 71 runs with three wickets remaining in their first innings.
Australia were 408 all out yesterday, after which Shikhar Dhawan blazed his way to becoming the fastest centurion on debut, leaving the Aussies demoralised.
But the visitors produced a much better effort today, picking up seven wickets in all.
Comfortably placed at 384 for three at lunch, India were taken by surprise after the break, with Australia staging a remarkable fightback to reduce the hosts to 431 for seven after Siddle had Ravichandran Ashwin caught behind.
Skipper Michael Clarke's decision to take the new ball seven overs after lunch paid dividends, as Starc had Vijay trapped with one that swung in before hitting the pads.
Bowling with a scrambled seam, Siddle induced an edge from Ravindra Jadeja to make it 427-6. The delivery moved away from the right-hander on landing and wicketkeeper Brad Haddin did the rest.
Ashwin followed six runs later, before Virat Kohli (50 not out), along with Bhuvneshwar Kumar, steadied the slide helping India score 95 runs in the second session.
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