Australia's pace bowlers had threatened to dominate day one of the series-opening test on Thursday.
The experienced number three did what he has done many times before and stood his ground as wickets tumbled around him, including superstar Virat Kohli for three, with India stumbling to 143-6 at tea on day one.
At the end of a day characterised by poor shot selection, Pujara went for a quick single on the second last ball and was removed in a brilliant run-out by a diving Cummins.
His knock inspired India to reach 250 for 9 at stumps on day one.
"We got the wickets the way we thought we might get the wickets", he told the Seven Network.
This is the first of the four-match Test series between India and Australia. Australian media, quoting police sources, said Arsalan Khawaja was being questioned about forgery and an attempt to pervert justice, but had never meant to commit a terrorism offense.
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The Test series is Australia's first on home soil since the ball-tampering scandal in South Africa and the hosts started strongly, with India slumping to 86-5 after lunch, before Pujara's resistance and contributions from Rohit Sharma (37), Rishabh Pant (25) and Ravi Ashwin (25) helped the tourists stage a fightback.
Opener KL Rahul, who has been struggling for runs, flopped again, caught by Aaron Finch at third slip off Hazlewood for two.
Veteran Murali Vijay, in the side after young gun Prithvi Shaw suffered ankle ligament damage, didn't last much longer, caught behind by skipper Tim Paine off Starc for 11.
Virat Kohli was dismissed for just three by a sensational catch from Usman Khawaja, before Cheteshwar Pujara struck a defiant century in an Adelaide Oval furnace to rescue India.
It will take some grab to top Khawaja for the catch of the summer. Rahane miscued yet another drive, this time to Handscomb at second slip.
Pujara then enjoyed 41-run and 62-run partnerships with Pant and Ashwin respectively before the 30-year-old, playing his 65th Test, was run out by Cummins, who along with Hazlewood and Starc each picked up two wickets. They slid further when Rohit Sharma holed out attempting a slog off Lyon.