India were dismissed 45 minutes into the morning session of day three for 157, adding just 16 to their overnight 141-6 with the irrepressible Scott Boland ending with 6-45 and 10 wickets for the match.
Set 162 to win, the hosts reached the target before tea against an attack missing injured strike bowler Jasprit Bumrah to regain the Border-Gavaskar trophy for the first time since 2014-15.
Australia lost three wickets before lunch to leave the Test finely-poised, including Steve Smith who agonisingly fell just one short of becoming only the 15th player and fourth Australian to reach 10,000 Test runs.
But Usman Khawaja hit a composed 41 before Travis Head (34 not out) and debutant Beau Webster (39 not out) steered them home.
Victory was enough to ensure they will defend their World Test Championship title in the June final at Lord's against South Africa.
It culminated a thrilling tour full of twists and turns that saw record crowds and some spectacular individual performances.
The five-Test series witnessed the emergence of exciting young talent in the form of Australian opener Sam Konstas and Indian all-rounder Nitish Kumar Reddy.
At the same time, ageing veterans, notably Rohit Sharma, showed that the clock is ticking on their careers.
India won the first match in Perth convincingly before Australia bounced back in the day-night affair at Adelaide.
The rain-hit Brisbane clash ended in a draw before the hosts took all the momentum into Sydney with a pulsating day-five win in Melbourne.
Indian pace spearhead and skipper Bumrah, who has been dealing with a back issue, batted briefly when his team resumed after bowling just one over and leaving the field on Saturday.
But in a huge setback the 31-year-old did not appear again, leaving Mohammed Siraj and Prasidh Krishna to lead the bowling attack, with Virat Kohli in charge.
Glimmer of hope
It appeared to be a difficult run chase on a deteriorating surface offering plenty for the bowlers, but Konstas and Khawaja had other ideas.
They raced to 35 off the first three overs, with teenager Konstas hitting out.
It proved his downfall, unnecessarily lashing at a good length ball from Prasidh only to sky it to Washington Sundar at mid-off, out for 22.
Prasidh then knocked over Marnus Labuschagne for six, caught by Yashasvi Jaiswal at slip, with the same pair removing Smith on four, fending off a rising ball.
Veteran Khawaja, who had a poor series, produced some cracking boundaries on his way to 41, but he mistimed a pull shot off Siraj with wicketkeeper Rishabh Pant collecting the edge.
It gave India a glimmer of hope before Head and Webster took the game away.
India resumed after a counter-attacking 61 from Pant helped build their second-innings lead.
Ravindra Jadeja began on eight and Sundar on six after they dismissed Australia for 181 in reply to their first-innings 185.
Jadeja crunched an early boundary off Pat Cummins, but he was out two balls later, getting a faint nick to wicketkeeper Alex Carey, adding just five to his overnight score.
The Australian skipper struck again to bowl Sundar for 12.
That brought Bumrah to the crease and he could only watch as Siraj edged Boland to Khawaja at slip to give the big Australian his fifth wicket.
Boland, whose line and length were impeccable all series, then bowled Bumrah to end the innings.
