Chasing 256 for win, Finch (110 not out) and Warner (128 not out) took the Indian bowlers to the cleaners to register the emphatic win with 74 balls to spare and take a 1-0 lead in the series.
Earlier, Shikhar Dhawan hit a half-century before Australia pulled things back to bowl India out for 255 after being invited to bat.
Dhawan looked in great touch during his 91-ball 74 and, in the company of K L Rahul (47), shared a second-wicket stand of 121 runs off 136 balls to set the base for India's total. However, India failed to capitalise.
When Dhawan and Rahul were at the crease, India were cruising towards a big total but their dismissal in space of 10 balls affected their run-rate. They raised a 121-run partnership for the second wicket.
The hosts struggled to raise any substantial partnership after that.
The second ODI of the series will be played in Rajkot on Friday.
India: 255 all out in 49.1 overs. (S Dhawan 74, KL Rahul 47; M Starc 3/56, P Cummins 2/44, K Richardson 2/43).
Australia: 258 for no loss in 37.4 overs (David Warner 128 not out, Aaron Finch 110 not out).