India captain Virat Kohli brought up the winning runs with his third boundary in his innings of 22 off 14 balls.
Opening batsman Shikhar Dhawan, who missed the one-day international series against the same opponents to be with his ailing wife, remained unbeaten on 15 off 12 deliveries.
The rain came down with Australia struggling on 118 for eight in 18.4 overs after being tied down by India's spin duo of Kuldeep Yadav and Yuzvendra Chahal.
Australia, who handed a debut to left-arm paceman Jason Behrendorff, were not able to complete their innings and India's target was adjusted.
Kohli won the toss and with the expectation of rain later, opted to field against the tourists, who had lost the five-match ODI series 4-1.
David Warner, captaining the side in the absence of the injured Steve Smith, fell in the first over after hitting two boundaries off seamer Bhuvneshwar Kumar but Australia seemed to have recovered through Aaron Finch and Glenn Maxwell.
Maxwell made 17 before once again falling to leg-spinner Chahal, who had dismissed him in all three matches he played during the ODI series.
Left-arm wrist spinner Yadav then dismissed the dangerous Finch for 42 and also all-rounder Moises Henriques, earning excellent figures of 2-16 off his four overs.
The second T20 will be played at Guwahati on Saturday.
(Reporting by Sudipto Ganguly in Mumbai; editing by Ian Chadband)
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