But the death knell sounded when Clint McKay (2-36) rattled the stumps of India skipper MS Dhoni (19) with 109 runs still required from nine overs, sending the bulk of the 34,745-strong crowd for the exits.
Shane Watson was also effective on the slow-paced wicket, grabbing 2-31, while ending Virat Kolhi's spirited knock of 61 off 85 balls.
Faulkner struck the early blow when he had Shikhar Dhawan (7) caught behind - but the aggressive allrounder celebrated a bit too vigorously and accidentally poked wicketkeeper Brad Haddin in the eye.
The Australian vice-captain had to leave the field and was off for 22 overs as he received treatment, but returned late to complete the game.
Earlier, Finch's aggressive 72 off 79 shot Australia off to a pacey start, before his skipper provided a more composed response.
He picking up the pace late to finish with 85 off 82.
Finch repeated his strong form from last Thursday's Twenty20 clash, again providing the backbone for the Australian innings.
The powerful right-hander blasted three sixes and eight fours in his 79-ball innings to propel Australia to an opening stand of 110 with Phil Hughes (47 off 52).
By contrast, Bailey kept the ball on the ground for much of his innings - hitting 10 boundaries - before being caught in the deep off Ravi Ashwin (2-55) when he attempted to put the ball into the crowd.
Glenn Maxwell (31), James Faulkner (27) and Clint McKay (11) added some lusty lower order hitting to push Australia's score past 300 in the dying overs.
- AAP
