India lost three wickets in the last two overs to reach a total of 104, and Australia chased down the modest total in 17.2 overs.
Meg Lanning led the way with a solid 39, studded with five boundaries, while Jess Cameron finished unbeaten on 36.
Lanning set the tone at the start with Alyssa Healy (23), as the duo put on 43 in under six overs. Lanning became the third player, male or female, to register five consecutive scores above 30 in T20 internationals.
Australia captain Jodie Fields was full of praise for Lanning’s performance, saying: "We expect her to set the tone at the top of the innings. She is fearless and extremely confident, I just love watching her bat."
India batted first and did well enough in the opening six overs, as Poonam Raut top-scored with a run-a-ball 21.
Later Mithali Raj and Harmanpreet Kaur added 25 runs for the third wicket. But the run-out of captain Mithali Raj proved to be India's undoing.
"Harman and I were batting really well, but we collapsed once again. One defeat doesn't mean the end of the road, we have to keep believing in ourselves. We can't sulk about this, surprises do happen," Raj said afterwards.
Women's World T20 fixtures:
Group B: New Zealand v South Africa (Galle) - Sep 28
Group B: Sri Lanka v West Indies (Galle) - Sep 28
Group A: Australia v Pakistan (Galle) - Sep 29
Group A: England v India (Galle) - Sep 29
Group B: South Africa v West Indies (Galle) - Sep 30
Group B: Sri Lanka v New Zealand (Galle) - Sep 30
Group A: India v Pakistan (Galle) - Oct 1
Group A: Australia v England (Galle) - Oct 1
1st Semi-Final: Colombo - Oct 4
2nd Semi-Final: Colombo - Oct 5
Final: Colombo - Oct 7
South Africa beat Sri Lanka by 6 wickets
West Indies beat New Zealand by 7 wickets
England beat Pakistan by 43 runs
Australia beat India by eight wickets