Finch had provided Australia’s best hope as he crashed 74 from 48 balls but suffered a double blow when he was run out and then pulled up injured after being called through for a quick single by Matthew Wade.
Finch dismissal came amongst a collapse of 63 for eight that helped India secure the three-match series before Sunday’s final match in Sydney.
Finch missed a large chunk of the county season last summer after requiring surgery on a hamstring injury suffered during last year’s Indian Premier League, which delayed his arrival at Yorkshire.
The right-hander will be desperate to avoid a similar lay-off, with Australia’s World T20 campaign due to begin on March 18, while the team is due to fly out for New Zealand next week for a limited-overs series.
India’s top-order has flourished ever since arriving in Australia and Sharma and Kohli continued their hot streak.
Sharma crashed 60 from 47 balls, and combined in a 97-run opening stand with Shikhar Dhawan, before Kohli ensured a testing target with 59 from 33.
Australia were on track at the halfway point of their pursuit, as Finch and Shaun Marsh racked up a 94-run stand, but India’s spinners then altered the course of the match as wickets constantly fell.