He finished with eight boundaries and five sixes in the whirlwind 35-ball knock - his highest international T20 score.
Singh put on 102 for the fifth wicket with skipper MS Dhoni (24 off 21), to steal the game from the tourists.
India required seven runs off the final over, bowled by Shane Watson, and Dhoni completed the task with two balls to spare.
Earlier, Finch, Glenn Maxwell and Nic Maddinson were primarily behind Australia's sixth highest T20 score.
Maxwell (27 off 13) and debutant Maddinson (34 off 16) provided entertaining cameos, but Finch again stole the show.
Just as he did with his world record 156 against England in August, Finch combined the brutal with the unorthodox to send India's attack to all parts of the boundary.
There were 14 fours and one six in a composed knock of 89 as Australia passed 200 for just the sixth time in T20s.
It took a spectacular caught-and-bowled from Vinay Kumar (3-25) to halt the powerful right-hander.
India's chase took a long time to gain traction, stumbling early when McKay (2-50) removed Rohit Sharma (8) in his first over.
Shikhar Dhawan (32 off 19) threatened to take the game away from Australia before he was expertly stumped by Brad Haddin after Xavier Doherty had lured him out of his crease.
Virat Kohli posed a similar danger but he, too, couldn't convert after making a start - holing out in the deep off McKay for 29 as India fell to 4-100 in the 12th over.
But the recovery fell to Singh and Dhoni, who had earlier won the toss and showed confidence in his team's ability to chase any target.
- AAP
