Rohit Sharma reacts as he leaves the field after being dismissed by Australia's Adam Zampa (left) during the final ODI match in Bangalore, India, on Sunday. AP
Sharma 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. Shreyas Iyer was unbeaten on 44.
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.
India fought back in the series after a 10-wicket drubbing in the opening game to outplay the Aaron Finch-led side in the final two ODIs. They won the second ODI by 36 runs.
Earlier, Steve Smith struck his first ODI century since his ball-tampering ban as Australia posted 286 for nine in the third and deciding match against India.
Smith made 131 and put on 127 runs for the third wicket with Marnus Labuschagne to lay the foundation for Australia's total after they elected to bat in Bangalore.
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 to 2019.
Indian bowlers checked Australia's surge in the end with paceman Mohammed Shami returning figures of 4-63.
He struck first to send David Warner trudging back to the pavilion for three after the left-handed opener edged to wicketkeeper KL Rahul.
Aaron 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.
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.
Labuschagne was caught at cover with skipper Virat Kohli taking a stunner while diving to his right, and three balls later Mitchell Starc, promoted to number five, was out for nought.
Bowlers hit back to rattle the opposition middle-order, but Smith kept up the charge from his end before falling to Shami.
Shami struck again in the over, bowling Pat Cummins for duck with a yorker.
India were dealt an early blow when Shikhar Dhawan injured his shoulder while fielding in the fifth over and stayed off the field. He was taken for an X-ray.
Indian players wore black armbands to honour the country's most economical bowler, Bapu Nadkarni, who died in Mumbai at the age of 86 on Friday.
India skipper Virat Kohli is back at the top of the batsmen’s list in the ICC Rankings for Test batsmen as Australia’s Steve Smith slipped behind him in the latest update
New Zealand, batting first in the decider, made 13 with Tim Seifert dropped twice before he was eventually dismissed. In reply, Indian captain Virat Kohli hit the winning runs with a ball to spare.
“Tendulkar batted in the toughest era of cricket. If he had to get the chance now, he may hit more than 1.30 lakh runs. So it’s not good to make a comparison between Sachin and Kohli,” Akhtar said during a live video session on Helo app.
The 34-year-old Spanish star also took aim at the international tennis schedule, which has been revised following disruptions sparked by COVID-19 shutdowns.
The 20-year-old Spaniard, City's first signing of the summer transfer window, has penned a five-year contract for a reported initial fee of £20.9 million ($27 million).
Ireland-born Morgan scored a century of his own as England set the visitors an imposing victory target of 329 at the Rose Bowl but he sat out the second innings because of a minor groin injury.