Rain delayed the start of play at the MCG on day five, but India quickly took the two wickets they needed to grab a 2-1 lead in the four-Test series.
Pat Cummins (63) and Nathan Lyon (7) were dismissed as Australia fell for 261, well short of their target of 399.
Jasprit Bumrah (3-53) had Cummins caught at first slip by Cheteshwar Pujara before Lyon edged behind off Ishant Sharma (2-40).
The fourth Test starts in Sydney on Thursday and India are on track to win a Test series in Australia for the first time.
After winning the toss and batting first, India’s 443-7 in the first innings proved to be decisive as the MCG pitch deteriorated.
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