Marsh produced one of his finest knocks for his country with 62 to steer Australia to 261-7, giving the hosts an overall lead of 326 runs heading into today’s final day.
The Indians will have to defy history at the famous Melbourne Cricket Ground (MCG), where they have not won for 33 years, to keep the Border-Gavaskar Trophy series alive to next week's fourth and final Test in Sydney.
The highest successful run chase in the fourth innings at the MCG remains 332-7 by England against Australia in 1928.
Teams have struggled to score more than 250 in the fourth innings on the drop-in pitch in recent years, with India all out for 169 (2011) and 161 (2007) in their last two Tests at the ground, both defeats.
Australia have to decide whether to declare overnight or bat on with a total of 96 overs to be bowled on the fifth day, but opener David Warner believes more runs are needed. "I think we need a few more runs than we've got at the moment,” he said.
The tourists kept the clamps on the Australian batting in a tense final session, before Marsh clawed back the advantage.
"Any score on the final day is going to be tricky," said India spinner Ravi Ashwin.
"It's going to be hard work, but we're up for it. We're here to win games of cricket, so we'll be positive and try and see where we can go."
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