Suryakumar Yadav (69 off 36 balls) and Virat Kohli (63 off 48 balls) guided India to a six-wicket win.
The win also saw India bag the series 2-1.
Earlier, Cameron Green (52) and Tim David (54) hit fifties to help Australia post 186/7.
For India, Axar Patel excelled with the ball, picking three wickets for 33 runs from four overs.
India vs Australia Live: India Win, Claim Series 2-1
A thick edge off Hardik's bat in the fifth ball of Daniel Sams' over and the ball raced to the boundary. That was all India needed as they recorded a six-wicket win over Australia and bag the series 2-1.
Earlier, Kohli (63 off 48 balls) started the over well for India dispatching Sams for a six in the first ball but lost his wicket in the second to send the whole crowd into a silence.
India vs Australia Live: India Require 11 Runs in Last Over
Good all-round display from the Australian fielders and bowler Hazlewood meant that India needed 11 of the last six to register a win. In the penultimate over, India could manage just 10 runs.3
India: 173/3 after 19 overs.
India vs Australia Live: Cummins Concedes 11 Runs
Australians made few mistakes on the field and Kohli and Hardik made good use of it by running between the wickets despite Cummins bowling a tight line. India require 21 off 12 after 18 overs.
India: 166/3 after 18 overs.
India vs Australia Live: Fine Over From Sams
Daniel Sams returned to the attack with India clearly on the frontfoot in the match. The Australian varied his pace and bowled a good length, giving away just seven runs.
India: 155/3 after 17 overs.