The Indians, coming out of coronavirus quarantine, looked rusty in their first taste of international cricket since February, as the Australians went on a run rampage to reach 374 for six after winning the toss.
Hazlewood then put the skids under the tourists, triggering a top-order wobble. Among his victims was the coveted wicket of skipper Virat Kohli, as India amassed 308 for eight in reply.
Australia enjoyed perfect batting conditions. Finch belted 114 and shared an opening stand of 156 with David Warner, who made 69 off 75 balls.
Smith had some early luck early before taking charge and hitting 105 off 66 balls with four sixes. He was saved by DRS after a confident appeal for leg before wicket on 15.
Then Shikhar Dhawan badly missed Smith's miscue on 38 at long on.
“I've been searching for something for a couple of months now, but a couple of days ago I started to hit the ball where I wanted it, and everything came back, the rhythm came back,” man-of-the-match Smith said.
“Finchy and Davey set an excellent platform, we could come and just take off.”
After Finch raised his 17th ODI century he was out for 114 off 124 balls trying to steer a Jasprit Bumrah lifter behind the wicket in the 40th over.
Glenn Maxwell hit a rumbustious 45 off 19 balls as Australia built a massive total to defend under the evening lights.
India made a fast start, the openers flaying the Aussie attack for 53 before Mayank Agarwal fell to Hazlewood for 22.
Kohli was dropped on one by Adam Zampa at fine leg after top-edging a Pat Cummins lifter.
The reprieve did not last long when the India skipper was cramped by Hazlewood and found Finch at mid-wicket, his night over at 21.
“We got enough time to prepare, I don't think there's any excuses on the field when you don't pull up as a side,” Kohli said.
“It's probably the first long game we've played in a while. The body language after 25-26 overs was disappointing.
“A quality side will hurt you [if you have fielding lapses]. We'll have to find out ways of getting a few overs from a few part-time [bowlers] in the side.”
Barefoot circle
Hazlewood picked up Shreyas Iyer, who failed to get his bat away from a bouncer and skying to Alex Carey for two, leaving the tourists at 80 for three.
Shikhar Dhawan and Hardik Pandya gave India an outside chance of pulling off victory in a 128-run partnership before Dhawan was caught by Mitchell Starc off Zampa for 74.
India's remaining hope went with Pandya who took on Zampa but failed to clear the rope and was caught by Starc for 90 off 76 balls. Zampa finished with four for 54 off his 10 overs.
India had not played international cricket since a tour of New Zealand earlier in the year and it showed with their outfielding ragged at times.
In contrast, Australia came into the match on the back of beating 50-over world champions England in September.
Even getting to the opening toss was an uphill battle for India, with wrangling over quarantine requirements and a recent outbreak of coronavirus in Melbourne complicating matters.
But Cricket Australia was desperate for the blockbuster tour, which also includes four Tests, to go ahead, with the funds generated crucial after the sport was hit hard by the Covid-19 shutdown.
Before play, both teams formed a ceremonial, barefoot circle to acknowledge the traditional indigenous owners of the SCG land.
Both teams also observed a minute's silence for the passing of Australia batting great Dean Jones, who died of a heart attack in India last September.
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