Chasing a formidable target of 359 runs, Peter Handscomb's maiden ODI century and half-centuries from Usman Khawaja and Ashton Turner took the visitors past the line with 13 balls to spare.
It was yet another superlative effort from the Indian skipper, who lacked adequate support from the other end as Australia kept the five-match series consequential going into the fourth game in Chandigarh on Sunday.
All-rounder Vijay Shankar scalped the last two Australian batsmen in the final over in which Australia needed to score 11 runs for a series-levelling win.
Maxwell's scintillating 113 off 55 balls overshadowed Virat Kohli's 38-ball 72 which helped India post a competitive 190/4 in 20 overs.
With 14 required off the last over, Umesh was hit for a boundary each by Jhye Richardson and Pat Cummins, who had identical scores of 7 not out as they reached the target of 127 off the last ball of the match.
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