On a high after back-to-back wins in Visakhapatnam and Thiruvananthapuram, India appeared keen to clinch the series in Guwahati as they racked up 222-3, their third successive 200-plus score of the series, riding Gaikwad's 57-ball blitz.
Maxwell, who bled 30 runs as a bowler in his only over of the match, had other ideas though.
Three weeks after smashing an unbeaten 201 against Afghanistan in the 50-overs World Cup, the all-rounder hammered an unbeaten 104 off 48 balls to secure a five-wicket victory for Australia.
Put into bat, India lost Yashasvi Jaiswal and Ishan Kishan inside three overs but stand-in skipper Suryakumar Yadav (39) combined in a 57-run stand with Gaikwad to arrest the slide.
Suryakumar fell at the midway stage of the innings but India reached 100 in 12 overs before Gaikwad stepped on gas.
He went run-a-ball for his first 21 runs before exploding with the bat.
Gaikwad clobbered seven sixes and 13 fours in what is the second highest score by an India batter in a T20 International, behind Shubman Gill's unbeaten 126 against New Zealand earlier this year.
He forged a 141-run partnership off just 59 balls with Tilak Varma, who made 31 not out. Sixty seven of those runs came in the last three overs.
Travis Head (35) led Australia's spirited reply before their top order wilted and the tourists slumped to 134-5 in the 14th over.
Maxwell turned the match on its head with his astonishing shotmaking, hitting eight sixes and as many fours.
Needing 21 runs off the last over from Prasidh Krishna, Australia captain Matthew Wade, who made 28 not out, took five runs from the first two balls getting Maxwell on strike.
Maxwell hit the third ball for a six, and followed it with a hat-trick of fours to seal a thrilling victory.
India's slow over rate meant they could deploy only four fielders in the deep for the final over, which proved costly in the end.
The teams move to Raipur for the fourth match on Friday, two days before the series concludes in Bengaluru.
