Virat Kohli proved to be the star man by scoring an unbeaten 82 to help the hosts to 161-4 in Mohali.
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The 27-year-old helped take 32 runs from the last 11 balls he faced as India chased down the Aussies' total of 160-6, allowing MS Dhoni to seal the win with five balls to spare.
Australia had reached 50 from 22 deliveries earlier in the day, but having been pegged right back it was left to Kohli to take over and blast India into the last four where they will meet the West Indies.
Elsewhere, England will face Group 2 winners New Zealand in Delhi on Wednesday for a place in the final.