The win also promoted Mahendra Singh Dhoni's boys to the top of the ICC T20 rankings and they now lead the table in the shortest and the longest format.
This is India's second highest run-chase after they overhauled 202 against Sri Lanka at Rajkot back in 2010.
Suresh Raina (49 no) and comeback man Yuvraj Singh (15 no) kept their cool as they blasted the required 17 runs off the final over to chase down a mammoth target of 198 set by Australia riding on stand-in captain Shane Watson's incredible 124 not out off 71 balls.
He had scratched his way to 5 off 9 balls before whipping a delivery on the pads over fine leg for a boundary.
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With 13 needed from five balls, it was vintage Yuvraj as he rocked back and smashed Tye over deep mid-wicket for a six to bring the equation down to 7 from four balls.
Yuvraj and Raina then scampered home for a single to make it 6 from three balls.
The UP left-hander kept his cool as he slashed it over point and the Indian dug-out erupted in wild celebrations as the players rushed to the ground to congratulate the duo.
Raina's 25-ball effort had half a dozen fours and a six.
However, a lot of credit should go to the in-form men Rohit Sharma (52, 38 balls) and Virat Kohli (50, 36 balls), who added 78 runs in 9.1 over.
Shikhar Dhawan's smashing cameo of 26 from 9 balls also helped India get off to a rollicking start.
(REOPENS FGN 8)
Dhawan and Sharma provided a blistering start to the Indian innings, taking 42 runs off the first three overs. Shaun Tait (0-46) and Scott Boland (0-34) were taken to the cleaners, particularly the former as he was taken for boundaries at will by both batsmen.
The run-rate shot up to 13-plus an over but Dhawan was the victim of his own aggression, caught behind off Shane Watson (1/30) in the 4th over.
They put on 78 runs for the 2nd wicket, with the runs coming off only 55 balls, gathering runs at pace as the Indian 100-mark came up in only 9.4 overs.
Sharma reached his 9th T20I fifty off 35 balls, in the 12th over then, but was out next straight up in the next over, caught by Watson at midwicket off leg-spinner Cameron Boyce (2-28).
Kohli meanwhile reached his 12th T20I fifty off 35 balls in the 15th over, and looked primed for the run-chase, when he deflected a delivery onto his stumps off the very next ball.
It left the innings in a lurch, with 51 more runs needed at that stage, when Yuvraj Singh (15*) walked in to bat for the first time in the series.
Indian bowling to smash an unbeaten 124 as Australia scored a commendable 197 for five in 20 overs after opting to bat.
The seasoned all-rounder played only 71 balls hitting 10 fours and half a dozen of sixes carrying his bat through the innings after winning the toss.
Such was Watson's dominance that the next best score was 26 from Travis Head.
Indian bowlers who were impressive in the first two games were belted out of the park as Australian batsmen hit a total of 16 boundaries and seven sixes in their innings.
Watson and Khawaja (14) then opened the innings against Ashish Nehra (1-32) and Jasprit Bumrah (1-43), and got off to a blistering start.
The duo punished the Indian pacers, especially Watson was severe on the youngster taking his two overs for 26 runs. Nehra sent back Khawaja in the 3rd over, caught behind, but Watson continued his onslaught on the bowling in the company of Shaun Marsh (9) as the Australian 50-mark came up in the 6th over.
It brought on a short passage of play as the spinners got on top, slowing the run-rate a bit. While Jadeja proved economical, Yuvraj had Glenn Maxwell (3) caught at covers in the 10th over, as Australia were placed at 80/3 at the half-way mark in the innings.
Watson then stepped on the accelerator, and in doing so brought up his fifty off 37 balls in the 11th over. Two overs later, he survived a life as Virat Kohli dropped him at deep cover off Hardik Pandya (0-24). He was unbeaten on 56 at that time.
It was also in that same over that Watson scored his maiden T20I hundred, off 60 balls, surpassing his previous best of 81. With three overs remaining in the innings then, Australia were placed at 168/4 and the batsmen then further upped the ante.
29 runs came off those overs then, as Watson remained unbeaten while Chris Lynn (13) was dismissed in the final over of the innings, caught in the deep by Jadeja off Bumrah. Cameron Bancroft (0*) was the unbeaten batsman.