With India chasing 188 and needing five runs off the last ball to win Devika Vaidya found the boundary to tie the match.
Smriti Mandhana and Richa Ghosh then smashed debutant Heather Graham for 20 runs, the most in a women's T20I super over.
Alyssa Healy, Ashleigh Gardner and Tahlia McGrath could manage only 16 runs in response as Australia's 21-match unbeaten streak was broken.
"What a game of cricket that was and what a showcase of fearless cricket that is over here in India and for thousands of people to be able to witness this is pretty special," said Healy.
"India were all over us right throughout that chase and the fact that we hung in there, took it to Super Over, we can be pretty proud of that."
It was a huge crowd for a women's match in India, and Australia's Graham was taken aback by the atmosphere.
"I'm not sure you can compare it (to anything else) to be honest," Graham said.
"I turned to Phoebe Litchfield who was debuting as well and I was just like, 'this is insane'.
"And we just soaked it all in. It's incredible to be able to come over and play in India and get a crowd like that, it's something I'm going to cherish forever."
