The win kept Sri Lanka unbeaten in four T20 internationals in Australia and followed their 2-1 series win over South Africa last month.
The pulsating win was made possible by a dashing 52 off 37 balls by Asela Gunaratne, who was named man-of-the-match for his match-turning innings.
Gunaratne slammed seven fours to share in a decisive 60-run stand with Milinda Siriwardana to put the Sri Lankans just 18 runs short of victory with 17 balls remaining.
But it was Kapugedera who stood up to the pressure and slammed fast bowler Andrew Tye's final delivery through the covers to the boundary rope for the winning hit.
The win put Sri Lanka one up in the series with two matches to play in Geelong on Sunday and Adelaide on Wednesday.
For a time it looked as though the Australians, playing without their leading stars, Steve Smith, David Warner, Mitchell Starc and Josh Hazlewood all on tour in India, would hold off the fast-finishing Sri Lankans.
Finch shared in an opening stand of 76 with T20I debutant Michael Klinger.
Klinger, making his debut at the age of 36 after starring with the Perth Scorchers in the domestic Big Bash League, hammered 38 off 32 balls with four boundaries.
Lasith Malinga, who has played virtually no cricket for nearly a year due to injuries, was an influential figure taking two wickets in two balls and taking two catches.
Malinga finished with two wickets for 29 off his four overs.
Paceman Pat Cummins struck with the last ball of his opening over getting skipper Upul Tharanga to edge to wicketkeeper Tim Paine for a duck in the first over of the Sri Lanka innings.
Leg-spinner Adam Zampa claimed the big wicket of free-scoring opener Niroshan Dickwella for 30 off 25 balls.
Dickwella, who slammed four fours and a six, looked to flick Zampa over mid-wicket only to lob a leading edge to Klinger at short third man in the ninth over.
But Gunaratne took charge and his half-century turned the tide to set up the thrilling victory for the Sri Lankans.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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