The home side had clinched the series with wins in Mohali and Indore, and a third successive defeat could potentially have deflated Australia ahead of their World Cup opener against tournament hosts India on October 8th.
Pat Cummins and his men did not let that happen, however, and racked up 352-7 with each of their top four batters smashing a half-century.
Playing his first match of the series, Maxwell proved he could be a potent spin option at the World Cup as he claimed 4-40 to help bowl out India for 286 with two balls left in the game.
Electing to bat, Australia's top order capitalised on a ground with a benign track and short boundary to lay the foundation for a big total.
David Warner smashed his third half-century of the series as he combined in a rollicking 78-run opening stand with Mitchell Marsh (96).
Warner fell for a belligerent 56 but Marsh added 137 runs with Steve Smith to consolidate Australia's position.
Kuldeep Yadav denied Marsh his hundred and had the opener caught at cover.
At 242-2 in the 32nd over, Australia looked to be cruising to a 400-plus total before the Indian bowlers dragged the hosts back into the contest.
Mohammed Siraj trapped Smith lbw for 74 to trigger a mini-collapse but Australia still registered the highest total at the venue with Marnus Labuschagne contributing 72.
Home captain Rohit Sharma (81), playing his first match of the series, led India's reply, smashing five sixes en route to a 31-ball fifty.
He and Virat Kohli (56) kept India on course until Maxwell intervened.
The off-spinner, playing his first international match since March, took a stunning return catch to dismiss Rohit and induced a top edge from Kohli to get rid of the former India captain.
Maxwell went on to peg back Shreyas Iyer's (48) off-stump to effectively seal the match for Australia and Ravindra Jadeja's breezy 35 was in vain.
