Electing to bowl after winning the toss, the Indians did well to restrict Australia to 141, before romping home in 18.4 overs.
In the process, the Indian women managed to chase down the biggest victory target against the mighty Aussies, who were the odds-on favourite for today's match.
The series now moves to Melbourne for the second game on Friday followed by game three in Sydney on Sunday.
However, debutant opener Beth Mooney's run-a-ball 36 followed by an entertaining knock of unbeaten 41 by Alyssa Healy lower down the order helped Australia post 140/5.
The Aussies were in a spot of bother before Healy's 15-ball blitzkrieg, which included five fours and a couple of sixes.
For India, leg-spinner Poonam Yadav was the top wicket-taker, scalping 2/26 in her 4 overs. Veteran pacer Jhulan Goswami, Shikha Pandey and Anuka Patil also picked up a wicket each in a consistently tight bowling performance.
However, the rest of the top-order batsmen did their job well enough with Player-of-the-Match Harmanpreet Kaur standing out with a 31-ball 46, which was laced with half a dozen fours and a six.
Smriti Mandhana (29) and Veda Krishnamurthy (36) were the other notable contributors in a fine chase.
For Australia, left-arm spinner Jess Jonassen grabbed two wickets for 24 runs in three overs.
After T20Is, the two teams will play a three-match ODI series early next month.
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