The in-form Delhi lad remained unbeaten on 82 runs off 51 balls as the Indians survived a few early setbacks to overhaul a difficult target of 161 with five deliveries to spare.
Sunday’s match, the last in Group 2, was a virtual quarter-final as the winners qualified the semi-finals. India finished their group engagements with three wins in four matches. Australia will go home with two wins and an equal number of defeats.
New Zealand had earlier qualified for the semi-finals from Group 2, while England and West Indies have gone through to the last-four stage from Group 1.
Shane Watson put in a superb all-round effort for Australia, finishing with figures of 2/23 in his four overs. Nathan Coulter-Nile (1/33) and James Faulkner (1/35) bagged a wicket each.
The Indians were off to a shaky start with openers Shikhar Dhawan and Rohit Sharma going back to the dugout without too many runs on the board. Watson snared Suresh Raina with a well-directed short-pitched delivery in the eighth over to leave the hosts in trouble at 49/3.
Kohli and Yuvraj Singh tried to script an Indian comeback, adding 45 runs between them in 38 balls. Yuvraj picked up a niggle in his left leg while attempting to glance a Coulter-Nile delivery to fine leg.
The experienced left-hander was clearly in some discomfort as he was hobbling while running between the wickets. But he battled bravely to post 21 runs off 18 balls with one boundary and a powerfully six off Adam Zampa.
But just as it seemed that Kohli and Yuvraj could take India to a safe position, the latter was undone by an excellent piece of fielding from Watson when he mistimed a slower one from Faulkner.
Yuvraj’s dismissal seemed to prod Kohli into action as he unleashed the big shots. The 27-year-old right-hander hit the hapless Faulkner for two boundaries and a six off successive balls in the 18th over to bring the target within India’s reach.
He then smashed four consecutive boundaries off Coulter-Nile in the next over to virtually wrap up the issue. With four runs needed in the final over, India captain Mahendra Singh Dhoni dispatched Faulkner’s low full toss to the long-on fence to complete a thrilling win for India.
Earlier, Australia posted a competitive total of 160/6 in their 20 overs.
Opener Aaron Finch was the highest scorer for Australia with 43 runs off 34 deliveries, hitting three boundaries and a couple sixes during his innings. Glenn Maxwell contributed 31 off 28 balls.
For India, all-rounder HardikPandya (2/36) copped some initial punishment before bagging a couple of crucial wickets while Yuvraj Singh (1/19), AshishNehra (1/20), RavichandranAshwin (1/31) and JaspritBumrah (1/32) also bagged a wicket each.
The Indians did not help their cause by conceeding as many as 14 extras.
Electing to bat first, Australia were off to an explosive start with openers UsmanKhawaja and Aaron Finch producing a 54-run partnership in just 26 balls. Bumrah copped a lot of punishment in his first over with Khawaja hitting him for four boundaries.
But Nehra gave India the breakthrough when Khawaja went after an outgoing delivery only to see the ball nick the top edge on the way to Dhoni behind the stumps.
The dangerous David Warner perished when he came down the track to Ashwin. The left-hander could not connect as the ball spun away from him and Dhoni pulled off an easy stumping.
Australia captain Steven Smith lasted for just six balls before Yuvraj had him caught behind with his very first ball. That saw the Australian run rate fall slightly with Finch and Maxwell struggled on a pitch which was offering a fair bit of turn.
Finch was deprived of what would have been a well-deserved half-century when he perished while trying to prop up the run rate. He tried to pull a slightly short-pitched delivery from Pandyainto the stands, but did not connect properly as the ball looped up for a fairly simple catch to ShikharDhawan at deep midwicket.
Maxwell was looking set for a big score. But he virtually gifted away his wicket to Bumrah when he went for a cross-batted heave only to be out-foxed by a slower delivery.
Peter Nevill and Watson hit a couple of fours and a six off Pandya in the last over to give Australia a strong finish.