After being asked to bat first, Australia posted a respectable total of 168-6, thanks to decent batting performances by openers Aaron Finch (43) and Michael Klinger (38).
First drop Travis Head also contributed significant 31 runs in their side's innings.
For Sri Lanka, Lasith Malinga scalped two wickets while Vikum Sanjaya, Asela Gunaratne and Lakshan Sandakan all chipped in with a wicket each.
In reply, Niroshan Dickwella (30) and Dilshan Munaweera (44) added 74 runs for the second wicket before Gunaratne smashed a quick-fire 52, including seven boundaries.
Milinda Siriwardana faced 11 balls to make 15 and was dismissed as Lanka's fifth wicket at 152 when the visitors needed 15 runs off 15 balls.
Chamara Kapugedara (unbeaten at 10) held his nerve till the end while Seekkuge Prasanna (unbeatn at eight) smashed the winning boundary when the game was tied in the last ball of the innings.
Lanka's final score read 172-5 in their stipulated 20 overs.
Adam Zampa and Ashton Turner bagged two wickets each while Pat Cummins also took a wicket.
Gunaratne was adjudged `Man of the Match` for his all-round performance.
Lanka, who have now taken a 1-0 lead in the series, will play their second T20 against Australia on Sunday at Shell Stadium in Victoria. (ANI)