Left-arm spinner Ravindra Jadeja then brightened India's victory prospects dismissing David Warner and nightwatchman Nathan Lyon cheaply to reduce Australia to 23-2.
Matt Renshaw was batting on seven at stumps with Australia still 129 runs behind and under considerable pressure to save the match.
"Obviously we've got to save the game," Australia coach Darren Lehmann told reporters, lamenting the two late losses.
"Once the ball gets a little bit softer it plays pretty well, so there's no real demons in the track.
"It's a case of obviously applying ourselves much like Pujara and Saha did today," he added.
Earlier, Pujara became the first Indian batsman to face 500 deliveries in a test innings, spending more than 11 hours at the crease for his marathon 202 on the penultimate day of the contest.
Resuming the day with India 91 runs behind Australia's first innings total of 451, Pujara and Saha frustrated the tourists and batted through the first two sessions to forge a 199-run seventh wicket partnership.
Both fell in the final session trying to accelerate with Australia looking pretty jaded by then.
Pujara faced 525 balls in his tireless knock, capitalising on the exceptionally fast outfield to score 21 boundaries at the Jharkhand State Cricket Association Stadium, which is hosting its first test.
Pujara was given out lbw to a sharply turning Nathan Lyon delivery when on 157 but he managed to prolong Australia's misery by using review to get the decision overturned.
The 29-year-old took a single off Lyon to bring up his third double century before falling to the off-spinner.
Saha matched Pujara's doggedness at the other end after surviving a couple of reviews earlier in the day.
Playing his second test more than five years after his first, Australia paceman Pat Cummins (4-106) nearly completed a five-wicket haul when Saha was given out leg-before.
The batsman immediately reviewed and replays suggested the ball would have missed the stumps.
Saha also got a reprieve on 51 when his fellow wicketkeeper Matthew Wade spilled an edge.
The stumper-batsman ducked well under the bouncers from Cummins and took a single off Glenn Maxwell to complete his century which drew his team mates in the dressing room on their feet.
Once past his hundred, he too went for quick runs and fell to Steve O'Keefe who bowled a staggering 77 overs for his three wickets as Australia endured a tough day in the field.
Down the order, Jadeja clobbered a couple of sixes in his brisk unbeaten 54 to rub salt into Australia's injuries, celebrating his fifty by twirling his bat like a sword.
Home skipper Virat Kohli called his batsmen in as soon as they crossed the 600-run mark and unleashed his spinners on Australia to drive home the advantage.
"I am very confident our spinners will bowl really well. There is rough, there are cracks..." Pujara told Star Sports channel.
The four-test series is tied 1-1 with Dharamsala hosting the final match from Saturday.
(Editing by Sudipto Ganguly)
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