Australia were dismissed for 131 with two balls left after winning the toss and India reached that target with 135 for two off 19.4 overs.
The Indians were always in control of the run chase, but tensions mounted in the final over as the Australians ringed the field around Gautam Gambhir before he pierced the circle for the winning runs with two balls to spare.
Gambhir remained unbeaten on 56 off 60 balls while skipper MS Dhoni, who promoted himself in the batting order to number four, was not out 21 off 18 balls.
Ravindra Jadeja was named man-of-the-match for his outstanding fielding effort of two run outs and a caught and bowled in Australia's innings.
India's win gives them some much-needed impetus in Sunday's opening match of the tri one-day series against Australia at the MCG also involving Sri Lanka in the tournament.
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