India were all out for 307 in their second innings of the first Test at Adelaide on Sunday, leaving Australia a record 323 to win.
Vice-captain Ajinkya Rahane smashed 70 and first innings century-maker Cheteshwar Pujara made a composed 71.
Three late wickets in the space of seven balls got the crowd on their feet and gave Australia hope of victory, but they will need to make history to win.
Spinner Nathan Lyon was the pick of the bowlers, taking 6 for 122 off 42 overs. It was his 13th five-wicket haul in Tests.
The only successful fourth innings Test run chase of more than 300 at Adelaide came in 1902, when the hosts made 315 to beat England.
The biggest on the ground in modern times was 239 in 1982, when the West Indies managed the feat.
A sliver of hope will come from Western Australia in a domestic match last month hitting 313 to beat South Australia here with Shaun Marsh -- who is playing in the Test -- scoring 163 not out.