For India, KL Rahul, who made 47 in the Indian innings, kept wickets after regular stumper Rishabh Pant suffered a concussion following a hit on the helmet while batting.
The manner of India's defeat, which came after dominating limited overs series wins against West Indies and Sri Lanka, has left the hosts with plenty to ponder ahead of the second game in Rajkot on Friday.
The last time India lost an ODI by 10 wickets was in 2005, against South Africa in Kolkata.
The Virat Kohli-led India were outclassed by Australia in the first ODI with a 10-wicket loss in first ODI at Wankhede Stadium, Mumbai.
Earlier, Dhawan hit a half-century before Australia pulled things back to bowl India out with five balls to spare.
"I think I've always had the hunger and desire for runs", Warner said after getting the man of the match award.
"It was a great effort from our bowlers in the afternoon".
He and Finch started the chase in belligerent fashion, putting on 84 runs in the first powerplay.
Of Australia's current touring squad, only Warner and Finch will be aged 35 or over at the next World Cup. The duo added 121 runs in less than 23 overs to take India to a strong position but they could not capitalise on that and lost seven wickets for less than 100 runs.
Warner took full advantage, cutting Jasprit Bumrah past point to bring up his 18th ODI century from 88 balls - as well as the 200 partnership - in the 31st over.
Kohli had just hit the spinner for a six but was dismissed brilliantly as the bowler took a sharp catch.
Finch reached his 16th hundred off 108 balls, including 13 fours and two sixes. Only Hashim Amla, Viv Richards and Virat Kohli have got their faster.
This was after Australia won the toss and opted to field. A few shots that Dhawan played went straight to the fielders while he also struggled initially to put bat to ball. He fell in the 28th over off Ashton Agar (1-56) and that's when India's collapse began.
With Kohli (16 off 14) dropping himself down the order, Shreyas Iyer (4), who had shown a lot of promise at number four in recent times, was demoted to five.
Cummins and fast bowler Kane Richardson picked up two wickets each for Australia. But a closer inspection would reveal where the Indians faltered and ended up with a sub-par total of 255.
"We were totally outplayed in all departments", Kohli said. "We got off to a flier, and we were very clinical in the end", he said.
"You lose games here and there, but this was one of the days where it didn't come off".
Zampa will head to Rajkot for Friday's second ODI with an advantage over the world's best batsman, having removed the Indian captain six times in global cricket.