Player of the Match Mitch Swepson (3-23) and Adam Zampa (1-21) did the damage after Australia posted 5-186, before restricting India to 7-174 in reply.
The Indians had already taken an unassailable 2-0 lead in the T20I series but Australia will take some joy in ending the visitors’ three-game winning run on the tour after triumphing in the final ODI.
“Very good to be back and good to be on the right side of the result,” Finch said.
“Over the three games it was a great series we were just on the wrong end on the first two.”
India captain Virat Kohli won the toss and sent the Aussies in, with Finch gone for a duck early, but Matthew Wade amassed 80 from 53 balls while Glenn Maxwell smashed 54 from 36.
Kohli led the charge in reply with 85 from 61 deliveries but was eventually dismissed by Andrew Tye as the pressure mounted on him with wickets falling around him.
Zampa and Swepson did most of the damage, with the former getting rid of Player of the Series Hardik Pandya for only 20. Swepson dismissed opener Shikhar Dhawan (28), Sanju Samson (10) and Shreyas Iyer (0).
“It’s the first time we’ve had two leg-spinners and that shows a lot of character from two guys to bowl to a really short boundary here,” Finch said.
“It shows a lot of courage, they backed their skills really well. I’m proud of them.”
Meanwhile, beaten skipper Kohli, who’ll only play the First Test of the upcoming series due to the impending birth of his first child, reflected on a positive white-ball series.
“This series win is a nice little asterisk for us finishing the limited overs leg of 2020 on a high,” he said.
“I honestly feel that the crowd was a factor as well. It’s great we can play in front of crowds in Australia. That gives you another dimension of motivation when you’re in a tough situation. Our fans pulled us through a couple of times and Australia has got great support too.”