The hosts reached 311-5 at stumps on the opening day with Tendulkar (109) putting on 146 for the fourth wicket with Venkatsai Laxman, who made a solid 64 in his 100th Test.
Sourav Ganguly, playing his final Test, was unbeaten on 27 with new captain Mahendra Singh Dhoni on four not out.
India looked in trouble when they slipped from 98 without loss to 116-3 in the morning session, with debutant off-spinner Jason Krejza (3-138) grabbing two wickets and seamer Shane Watson one.
But Tendulkar, 35, and Laxman ensured their team did not lose the advantage of winning the toss on good batting pitch. Both scored freely against pace and spin.
India lead 1-0 in the series following their 320-run victory in the second Test in Mohali. The matches in Bangalore and New Delhi were drawn.
Australia, needing a win here to retain the Border-Gavaskar Trophy, failed to maintain pressure on the batsmen.
Pacemen Brett Lee (0-46) and Mitchell Johnson (1-54) bowled well only in patches, while leg-spinner Cameron White failed to test the batsmen on a first-day wicket. Krejza's three wickets proved expensive.
Salute to prince of Indian cricket
Australia also failed to make the most of the opportunities that came their way, with Johnson dropping Tendulkar on 85 off Krejza at deep mid-off when the batsman stepped down the track to attempt a big shot.
The Indian batsman offered another chance on 96. Lee ran back from mid-off to hold a skier but failed to judge it properly with Krejza again the unlucky bowler.
Barring these lapses, Tendulkar looked in command and executed handsome shots on both sides of the wicket. He hit 12 fours in his 188-ball knock before being trapped leg-before by Johnson, bowling with the second new ball.
India were off to a brisk start, with Virender Sehwag (66) dominating a 98-run stand for the opening wicket with Murali Vijay who made an impressive 33 on his debut.
Very very special occasion for Laxman'
Australia's first success came when Watson had Vijay caught behind off a lifting delivery. Krejza then took two wickets in three overs, dismissing Sehwag and Rahul Dravid for a duck.
The spinner, hit for a four and a six by Sehwag in his opening over, extended Dravid's poor run with the bat when he had the Indian caught by Simon Katich at short-leg with a delivery that turned and bounced.
Dravid has scored just one half-century in the series.
Krejza, 25, celebrated again when he removed Sehwag, bowled off an inside-edge while attempting to cut. Sehwag hit one six and nine fours in a 69-ball knock -- his 16th Test half-century.
His third wicket came in the last session when he had the well-set Laxman caught behind.