But needing a win to level the four-match series, Australia crumbled to 300 all out, compounding the misery after some soft dismissals by the top order on Saturday as they stare down the barrel of defeat.
India's 2-1 series win was confirmed after the final day of the fourth Test was washed out and the match ended in a draw.
Kohli's team banked wins in Adelaide and Melbourne to take an unassailable 2-1 series lead and retain the Border-Gavaskar trophy, while they took control of the final Test by amassing 7 (dec)-622 after winning the toss.
Cheteshwar Pujara was all praise for the bowlers. We've had four bowlers, and it's not easy taking 20 wickets, so credit goes to all our fast bowlers and spinners, remarkable.
Asked about his tremendous form in the Test series, Pujara said, "I am really pleased with my contribution".
Play at the Sydney Cricket Ground had finally got underway after a rain delay of nearly four hours and India needed just 80 minutes to take the last four wickets at a cost of 64 runs.
"I think what we've been able to achieve is something we should be very proud of", Kohli added.
"We've definitely had opportunities to make hundreds in the series, it's been execution or a few other amusing ways that people have got out, so that's it really".
"I have only one word to say - that I'm proud, firstly as part of the team, then to lead these players which is an honour and a privilege".
"The next Test series is some 6-7 months away, it'll give me some time to prepare".
"We've been outplayed in some areas", Tim Paine conceded to Fox Cricket prior to the start of play on Monday. Australia definitely had their moments, but India dominated major parts of the series and deserved the historic win.
"Have to tip our hat to India, we know how tough it is to win in India so congrats to Virat and Ravi because it's a huge achievement".
The two sides now play a five match one day series before India, who're the top ranked one day side in the world, head to New Zealand for a five match series against the Black Caps.
If Australia can't beat Sri Lanka at home, god help us in The Ashes in England later this year - with or without the returning Steve Smith and Dave Warner. Leave alone a three-figure score, none of the Australian batsman could even cross the 80-run barrier, something that legends like Ricky Ponting, Shane Warne, Mike Hussey, Ian Chappell found hard to fathom. We know there are guys outside of this group who want to come in.