The win helped the Indians take unassailable 2-0 lead in the four-match series.
Spinners R Ashwin and Ravindra Jadeja made the most of the turning track at the Rajiv Gandhi International stadium, taking eight wickets apiece and bowling out Australia for a meagre 131 in 67 overs in their second innings.
The visitors needed 192 runs to avoid an innings defeat but they lost the final eight wickets in 35 overs with an addition of only 57 runs to their total.
Cheteshwar Pujara was given the man-of-the-match award for his double century.
The third Test between the teams will be played in Mohali after an eight-day break.
The fact that Australia lost their last three wickets for an addition of only one run in 10 overs shows their sheer inability to face spin bowling on Asian wickets.
With this win, Mahendra Singh Dhoni became the most successful Indian captain, eclipsing Sourav Ganguly’s record of 21 Test wins.