Sharma made 119 and put on a key 137-run second-wicket stand with skipper Virat Kohli, who made 89, as India chased down their target of 287 in 47.3 overs in Bangalore.
The hosts fought back in the series after a 10-wicket drubbing in the opening game to go past the Aaron Finch-led tourists 2-1.
"We lost the last three in the last series (in 2019), and coming back and winning the last two games after losing the first one, is very satisfying," said Kohli, whose side lost to Australia 2-3 at home last year.
"2020 has been wonderful, and we just want to go onwards and upwards. Great for the team."
Paceman Mohammed Shami claimed four wickets to lead the bowling charge with the hosts restricting Australia to 286 for nine on a good batting pitch despite Steve Smith's 131.
Sharma took the attack to the opposition with a steady flow of boundaries -- eight fours and six sixes in his 128-ball knock -- and completed his eighth century against the Aussies.
He finally fell to Adam Zampa's leg spin as the senior batsman attempted a big hit but the ball top-edged off his bat to be caught by Mitchell Starc at long-on.
Kohli kept the chase on track with an attacking Shreyas Iyer, unbeaten on 44, as the pair put on 68 runs before the captain was bowled by Josh Hazlewood.
Indian opener Shikhar Dhawan did not bat after injuring his shoulder while fielding in the fifth over Australian innings. He went for an X-ray later and TV pictures showed the batsman's arm in a sling.
India though did not feel his absence as Sharma and wicketkeeper-batsman KL Rahul put on 69 runs to set up the chase.
Rahul was given not out by the on-field umpire after being trapped lbw off Ashton Agar for 19 but Australia successfully reviewed the decision in their favour.
Smith century and 'great learning'
Earlier Smith followed up his 98 in the second match with his ninth ODI hundred and his first since January 2017 -- a period during which he served a one-year ban between March 2018 and 2019.
Smith put on 127 runs for the third wicket with Marnus Labuschagne to lay the foundation for Australia's total after they elected to bat.
Shami struck first to send David Warner trudging back to the pavilion for three after the left-handed opener edged to wicketkeeper KL Rahul.
Finch and Smith started to build, but their partnership was cut short by a big mix up between the two batsmen with the captain being run out for 19.
Smith played the ball to short third man and Finch ran for a single only to get stuck with his partner at the striker's end, and fell well short of the crease while trying to make it back.
Smith kept the scoreboard ticking with singles and twos and the occasional boundary with partner Labuschagne, who got a maiden ODI fifty after making his debut in the opening match of the series.
Left-arm spinner Ravindra Jadeja broke the partnership with the wicket of Labuschagne for 54, and struck twice in his ninth over to pull Australia back.
Smith played a lone hand before falling to Shami as the middle and lower order did not fire.
"We probably lost just one wicket (too many)," said Finch.
"Just that momentum kept getting stalled every now and then. To play against the best in the world in these conditions is a great learning experience."
India now head to New Zealand for a full series starting with five Twenty20 matches in Auckland on January 24.