The visitors were bowled out for 355 in reply to India's first-innings total of 441 on the third day. With just two days to go, they will need an extraordinary effort to win the match and square the series.
Australia trail 1-0 after their 320-run defeat in the second Test in Mohali. The matches in Bangalore and New Delhi were drawn.
The hosts could not open their account in the second innings after facing just one over. Virender Sehwag and debutant Murali Vijay were at the crease.
India's new captain Mahendra Singh Dhoni succeeded in his bid to strangle Australia with a defensive approach as the visitors found runs hard to come by on a dull day.
Australia, looking to score briskly after racing to 189 off 49 overs on Friday, managed just 166 off 85.4 overs as the hosts set a defensive field in the first two sessions.
Fast bowlers Zaheer Khan and Ishant Sharma were instrumental in drying up runs as they hardly made the batsmen play by bowling outside the off-stump, especially in the morning.
Off-spinner Harbhajan Singh was the most successful bowler with 3-94. Sharma, who saw two catches dropped off his bowling, finished with 2-64 and leg-spinner Amit Mishra ended with 2-58.
India gained the upper hand when they grabbed three big wickets in the space of 11 runs in the afternoon, dismissing well-set Michael Hussey (90), Michael Clarke (eight) and Shane Watson (two).
The visitors were comfortably placed at 229-2 following a 155-run stand for the third wicket between left-handed opener Simon Katich (102) and Hussey before losing their way.
Zaheer provided the breakthrough after Australia resumed at 189-2, trapping Katich leg-before with a swinging yorker. The Australian hit nine fours in his 189-ball knock.
Hussey looked set to complete his 10th century in 29 Tests before being run out following a smart piece of fielding by Vijay at silly point.
He pushed Harbhajan and set off for a single, but Vijay stopped the ball and flicked it to wicket-keeper Dhoni who whipped off the bails, with the batsman out of his crease. Hussey hit eight fours in his 228-ball knock.
Sharma found the edge of Clarke's bat with a gem of a delivery that moved away after pitching on the off-stump, with Dhoni taking an easy catch.
Watson fell playing a defensive shot off Harbhajan, the ball rolling back to hit the off-stump.
Cameron White made 46 and Brad Haddin 28 late in the innings, but their efforts were not good enough to help their team gain the lead.
Katich, 92 overnight, reached his fifth century in 30 Tests in the opening hour when he steered Sharma to third-man for his ninth four. He gave a chance on 94 off Sharma, but Rahul Dravid dropped a regulation slip catch.
Hussey, 45 overnight, reached his 12th half-century, and third of the series, with an edged boundary off Zaheer.