The deluge cut off a road, uprooted trees, washed away homes and sent vehicles flying
Warner blasted 128 while captain Finch made 110 as Australia chased down their target of 256 in just 37.4 overs in Mumbai to take a 1-0 lead in the three-match series.
Paceman Mitchell Starc also played a key part with three wickets that helped bowl out India for 255 in 49.1 overs after the tourists elected to field first in their first 50-over outing since last year's World Cup.
Warner and Finch then put together an unbroken stand of 258, the highest partnership for any wicket in ODIs against India.
"I think I've always had the hunger and desire for runs," Warner said after getting the man of the match award.
"It's just a great effort from our fast bowlers to knock India over for 255... We got off to a flier (while chasing), and we were very clinical in the end."
Warner reached his 18th ODI hundred in 88 balls with a boundary off Indian quick Jasprit Bumrah, jumping with joy and kissing his helmet to celebrate the knock.
Finch completed his century in 108 balls but it was Warner who remained in total control as he finished it off with two of 17 boundaries. He also hit three sixes.
KL Rahul, who made 47 for India, kept wicket after regular gloveman Rishabh Pant suffered a concussion following a blow on the helmet while batting.
It was the second-wicket 121-run stand between Rahul and Dhawan that put India on course for a big score but they slipped from 134-1 to 164-5 in the space of 35 deliveries.
The left-handed Dhawan smashed nine fours and one six in his 91-ball knock and was well supported by Rahul, who eventually fell to Ashton Agar's left-arm spin.
Pat Cummins was brought back from the other end and sent Dhawan back to the pavilion after he flicked to cover where Agar took the catch.
Skipper Virat Kohli did not last long, hitting Adam Zampa for a six before he was caught and bowled with the next ball for 16.
Pant and Ravindra Jadeja, who hit 25, made 49 runs with an attacking partnership. But Kane Richardson claimed Jadeja and Cummins got Pant for 28.
A short ball from Cummins hurried onto Pant who went for a pull and got a top edge onto the helmet, the ball lobbing towards point.
Number 10 batsman Kuldeep Yadav made a useful 17 to take India's score past 250 before being run out by a direct throw from Steve Smith.
"It's a very strong Australian team, and if you don't play well enough, they will hurt you," said Kohli.
"In phases, we were too respectful against their bowlers, and didn't take the game by the scruff of the neck. It's another challenge to bounce back from here."
The second ODI is on Friday in Rajkot.Scoreboard
India
R. Sharma c Warner b Starc 10
S. Dhawan c Agar b Cummins 74
KL Rahul c Smith b Agar 47
V. Kohli c & b Zampa 16
S. Iyer c Carey b Starc 4
R. Pant c Turner b Cummins 28
R. Jadeja c Carey b Richardson 25
S. Thakur b Starc 13
M. Shami c Carey b Richardson 10
K. Yadav run out (Smith) 17
J. Bumrah not out 0
Extras (lb3, w8) 11
Total (all out, 49.1 overs) 255
Fall of wickets: 1-13, 2-134, 3-140, 4-156, 5-164, 6-213, 7-217, 8-229, 9-255, 10-255
Bowling: Starc 10-0-56-3 (2w), Cummins 10-1-44-2 (2w), Richardson 9.1-0-43-2 (2w), Zampa 10-0-53-1 (1w), Agar 10-1-56-1 (1w)
Australia (target 256)
D. Warner not out 128
A. Finch not out 110
Extras (b4, lb7, w9) 20
Total (0 wkts, 37.4 overs) 258
Did not bat: S. Smith, M. Labuschagne, A. Turner, A. Carey, A. Agar, P. Cummins, M. Starc, K. Richardson, A. Zampa
Bowling: Shami 7.4-0-58-0 (1w), Bumrah 7-0-50-0 (1w), Thakur 5-0-43-0 (4w), 10-0-55-0 (2w), Jadeja 8-0-41-0 (1w)
Toss: Australia
Result: Australia won by 10 wickets
Series: Australia lead the three-match series 1-0
Umpires: Michael Gough (ENG), Chettithody Shamshuddin (IND)
TV umpire: Richard Kettleborough (ENG)
Match referee: Ranjan Madugalle (SRI)
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