The hosts posted 6-186 on Sunday night on the back of an explosive 74 from 38 balls from Tim David batting at No.4 and Marcus Stoinis half-century.
Australia kept themselves in the hunt with regular wickets in India's chase, before Washington Sundar's unbeaten 49 from 23 got India home.
After a washout in the series opener in Canberra, India's win puts the five-game series at 1-1 with two fixtures remaining.
On his Hurricanes home turf, quick Nathan Ellis did his best to inspire Australia with 3-36, including the scalps of openers Abhishek Sharma (25) and Shubman Gill (15).
But India were able to keep on top of the run-rate to set up a crucial game four on the Gold Coast on Thursday.
Australia missed elite quick Josh Hazelwood, who took 3-13 in game two but is sitting out the final three fixtures of the series with the Ashes looming.
Earlier, after Australia were sent in, David whacked an impressive 74 from 38 balls featuring five sixes and eight fours.
He came to the crease with his side 2-14 in the third over and made India pay for dropping him on 20.
David launched two sixes from the spin of Axar Patel in the seventh over, including one straight that hit the signage on top of the Ninja Stadium roof.
He fell in the 13th over, caught on the long-off boundary off the medium pace of Shivan Dube.
Stoinis held Australia together somewhat with an important 64 from 39 balls, in an innings in which three of the top five fell for single digits.
India spinner Varun Chakravarthy (2-33) suffered some punishment from David, but took two scalps in two balls including rattling Mitch Owen's stumps for a duck.
An inclusion after Melbourne, left-arm quick Arshdeep Singh was the pick of India's bowlers with 3-35 and won man-of-the-match honours.
India's chase got off to a strong start before twin blows to Ellis had them 2-62 in the sixth over.
The tourists lost skipper Suryakumar Yadav for an 11-ball 24 when he chipped a catch to Ellis off the bowling of Marcus Stoinis in the eighth over.
Sundar was immediately aggressive and to took the long handle to Sean Abbott with back-to-back sixes.
Australia could turn to Glenn Maxwell in game four. The allrounder has been given slightly longer to recover from a wrist injury after joining the squad for games three to five.
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