Chasing a victory target of 273 runs, the hosts were all out for 237 runs in the stipulated 50 overs.
Coach Justin Langer also praised Australia's "unbelievable fight and great character", while India skipper Virat Kohli credited the opposition for showing more "passion, hunger and heart than us in the last three games".
Bhuvneshwar Kumar claimed three wickets while fellow paceman Mohammed Shami and Ravindra Jadeja took two each. Usman Khawaja compiled a 106-ball 100-run knock - his 2nd ODI century - while it was leg spinner Adam Zampa, in association with Pat Cummins, that deflated the Indian batting line-up, which for some odd reason was an experimental one at best.
Seldom in their cricketing history Australia have walked in to a series as the underdogs, but they walk out with the heads held high, inflicting a series defeat on the World No.2 side in their final ODI series before the 2019 ICC Cricket World Cup.
Australia travel to Dubai on Thursday for a five-match ODI series against Pakistan, which will be the final chance for players to impress before selectors name a 15-man World Cup squad. However, Rohit made a solid comeback during the fourth ODI, where he hammered 95 runs when the team needed the most.
Sharma and Kohli added 53 runs for the second wicket at quick pace.
Australia claims historic one-day international series win in India
He smashed leg-spinner Zampa for a six but was undone by Lyon, who had the left-hander caught at first slip. India needs 51 off 27 balls.
The extravagance cost both Pant and Vijay Shankar (16) their wickets as they went after Lyon and Zampa respectively.
After winning the toss, Australia were limited to 272/9 - after Khwaja, Finch and Peter Handscomb had helped the team to a formidable 175/1 in 32.5 overs - losing their last eight wickets for just 97 runs.
Khawaja took a single off Yadav to bring up his hundred but fell soon after to Bhuvneshwar Kumar (3 for 48), triggering a mid-innings collapse for Australia who had been cruising at 175 for 1 in the 33rd over.
The left-handed batsman - with scores of 50, 38, 104 and 91 in the previous matches - paced his innings smoothly during his knock that was laced with 10 fours and two sixes. Khawaja drove an outside off stump delivery straight to the hands of Kohli at short extra cover.
It turned out to be a forgettable day for chinaman Kuldeep Yadav, who bled 74 runs in his 10 overs, although he dismissed risky Ashton Turner (20) cheaply.More news: Vietnam asks Malaysia to free Kim assassin suspect
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