The Indians, propelled by vice-captain Kohli, emphatically took a 1-0 lead in the three-game series ahead of matches in Melbourne on Friday and Sydney on Sunday.
Kohli pounded nine fours and two sixes in his knock, punishing Australia's fast bowlers.
Express quick Shaun Tait conceded 45 runs in four wicketless overs which included six wides and fellow pacemen James Faulkner (1-43) and Kane Richardson (0-41) were also expensive.
Only evergreen Shane Watson, who thought his international days had ended when he last played for Australia last September, stemmed the flow.
Watson finished with 2-24 from four overs, striking with his first ball and again four balls later.
But Australia couldn't contain Kohli, who reached his half-century from 32 balls and featured in a telling partnership with Suresh Raina (41 from 34 deliveries).
The pair put on a 100 runs from 68 balls en route to a 132-run stand which was only broken in the final over.
Australia's reply with the bat began soundly with Finch and David Warner taking 47 runs from the initial five overs before the latter fell for 17.
Finch and Smith then looked promising in a 42-run union but Smith's dismissal to Jadeja started the home side's slide.
Australian skipper Finch, despite top-scoring, said he largely blamed himself for the loss because he didn't kick-on in his innings.
"I allowed us to get too far behind the rate from the start," Finch said.
"Davey Warner and Steve Smith were outstanding with their intent and I just couldn't get off strike, I couldn't hit boundaries for a period there so I take full responsibility for them getting out."
Indian captain MS Dhoni praised the feats of his spinners to curtail the Australian run chase.
"We have seen spinners be quite successful in this format," said Dhoni, who also hailed the intelligence of Kohli's well-paced innings.
