Australia debutant Moises Henriques was left stranded on 81 as his side were bowled out for 241, leaving India chasing just 50 to win the first Test.
That simply was not enough to cause any jitters for India, even when the wickets of Murali Vijay and Virender Sehwag fell to James Pattinson and Nathan Lyon, and there was time for Sachin Tendulkar to delight the home crowd with maximums off his first two deliveries.
Vijay perished for six - all of which came via a straight, flat maximum down the ground off Lyon - as he picked out Henriques at short mid-off when he mistimed a drive off Pattinson.
Sehwag followed in unusually tame fashion, edging to Michael Clarke at first slip as he defended the spinner.
Few in the Chennai crowd were disappointed, though, as that brought Tendulkar to the crease - and he gave them what they wanted by heaving his first two balls off Lyon for successive sixes.
He could not make it a hat-trick to win the game, though, but an edge for a single from Cheteshwar Pujara did the job in any case.
Earlier, Lyon could not help Henriques reach a debut century as Australia added just nine to their overnight score.
Lyon's stout defence helped to repel the Indian bowlers for around 20 minutes before he was finally caught by Vijay at short-leg off Ravindra Jadeja.
The second of four Tests begins on Saturday in Hyderabad.
- Sports & Recreation
- Moises Henriques
- Murali Vijay
- Sachin Tendulkar
- James Pattinson
- Virender Sehwag