However, Handscomb and Marsh absorbed the pressure with their calculated knocks to deny the hosts, who apart from the sizeable crowd, were also cheered on by former captain and local favourite Mahendra Singh Dhoni.
India were all over the Australian batsmen in the morning session of Day Five, especially after skipper Steve Smith was dismissed by Ravindra Jadeja.
"Credit to the groundsman, the wicket held together beautifully".
He had a half chance on six when Karun Nair had a hard chance at short leg off Ravi Ashwin, but otherwise he defended confidently. "We were happy that we got ourselves into a position where we could play for a win".
The tourists began day five on 23-2, still trailing India by 129, after the hosts declared on 603-9 in response to Australia's first innings 451.
Pujara scored 202, spent over 11 hours at the crease and faced a record-breaking 525 deliveries.
At the end of the day, India were at 360/6 in 130 overs.
Matt Renshaw will restart on seven not out, with captain Steve Smith due to join him.
When Pujara and Saha were batting well in tandem, the Aussies needed a spark of brilliance on the field to lift themselves up in tough conditions.
Australia lost just four wickets on the final day on a pitch that was far from unplayable despite the ominous signs late on the fourth day. Virat Kohli celebrated both wickets heartily, grabbing his shoulder in a clear sign he didn't appreciate Glenn Maxwell's antics on the preceding day.
Pujara was the beneficiary of a marginal lbw call, prompting Australia to review unsuccessfully as Steven O'Keefe looked to have beaten the inside edge of the bat, only for replays to prove inconclusive before the original decision was upheld.
Australia took the second new ball immediately after 80 overs and Cummins earned them the most prized wicket of Kohli.
O'Keefe, who created the straightforward chance that Wade put down in the 156th over, bowled 77 overs.
Australia were comprehensively outplayed in the second half of the match but displayed a level of resilience seldom seen by the men in baggy green in this part of the world.
Playing his second test more than five years after his first, Australia paceman Pat Cummins (4-89) almost completed a five-wicket haul when Saha was given out leg-before.
Left-arm spinner Jadeja relentlessly attacked the rough outside Renshaw's off-stump, while Ishant Sharma bowled to Smith with a seven-two off-side field. Kohli, who went off the field on the first day due to an injury and did not field the whole of second day yesterday, had scored 0, 13, 12, 15 in the four innings of the earlier two Tests. Opener KL Rahul contributed with solid 67 runs before being caught on the bowling of Cummins.