Cheteshwar Pujara produced yet another classy hundred and stitched a 170-run stand with Virat Kohli (82) to put India in command against Australia.
Kohli contributed a significant 82 off 204 balls, made a surprising declaration and finished India's first innings at 443 for 7.
Rohit Sharma ended on 63 not out off 114 balls, inclusive of five fours, while Pat Cummins was the pick of the Australian bowlers, taking 3/72 in a back-breaking effort of 34 overs.
Rahane got unlucky shortly afterwards though as one delivery from Lyon kept too low and he was plumb lbw.
At stumps, Australia were placed at 8 for no loss from 6 overs with openers Aaron Finch and Marcus Harris were both unbeaten on 3 and 5 respectively.
Rishabh Pant (39) played an uncharacteristic defensive inning, and the only aerial stroke he played was in the 159th over off Lyon.