The pool clash was essentially a quarter-final, with the winners facing West Indies in Thursday's semi-final at Mumbai and the losers eliminated from the tournament.
The sold-out crowd was heaving, even by Indian standards, and James Faulkner had the ball in hand.
Kohli belted Faulkner all over the park, reducing the victory equation to 20 off the final two overs.
The gifted batsman attacked Nathan Coulter-Nile with similar success the following over; sinking to his knees when MS Dhoni struck the winning runs.
"This innings has to be in my top three, perhaps the top right now because I'm a bit emotional," Kohli said.
Skippers Smith and Dhoni were both full of praise.
"It was the Virat show," Smith said.
"It just seemed like anything Virat was going to do at that point was going to go to the boundary.
"We were in a pretty good position at one point ... it took a seriously unbelievable innings under pressure from Virat Kohli to get his team over the line.
"He timed his chase to perfection and he's done that so consistently in T20 cricket. "Hats off."
Kohli nailed the tricky run chase as teammates folded.
Shane Watson starred in the topsy-turvy tussle, winding the clock back in his final game for Australia.
Watson removed Rohit Sharma and Suresh Raina in a haul of 2-23, while he claimed a scrambling one-handed catch to dismiss hobbled veteran Yuvraj Singh.
But Australia couldn't break the partnership that mattered most.
"It was an incredible innings. Especially the fact that the wicket was not easy to bat on," Dhoni said. "He's batting amazingly well."
The 27-year-old seems to save his best for Australia and this dig was no exception, he rowed with Glenn Maxwell after taking exception with a return but never lost his cool. Kohli picked gaps no matter what fields Smith set, turned ones into twos and stroked 11 boundaries in a man-of-the-match effort.
Australia were denied by a superstar who averages 122.83 in successful T20 chases for India but they should have set a more imposing target.
Usman Khawaja and Aaron Finch blitzed the opening stanza, bringing up their 50-run partnership in just 3.4 overs after Smith won the toss.
Momentum shifted dramatically after Khawaja was caught behind.
Smith, David Warner and Aaron Finch, who top-scored with 43, all fell in a slump of 3-28. "We probably did let ourselves down in the middle overs again," Smith said.
The majority of Australia's 15-man squad will remain in India for the Indian Premier League, which starts next week.
-AAP
