Chasing 169 for victory, the tourists needed six to win off the final over bowled by paceman Andrew Tye. Scores were locked after five balls but, with the field called in, Chamara Kapugedera was able to find a gap and slap a boundary through cover to post an exciting victory.
Australian captain Aaron Finch praised Sri Lanka's bold batting.
"I thought the way they attacked the chase was outstanding - credit to them, they batted beautifully," he said.
The Australians missed the star power of Steve Smith, David Warner, Glenn Maxwell and Mitchell Starc, who are all in India preparing for a Test series, beginning on Thursday.
Skipper Smith posted a century against India A in Mumbai on Friday, but his talents would have been more than appreciated before a strong crowd of 42,511 at the MCG, who clearly had no concerns about coming to see what some had labelled a second-string Australian side.
However, players have already spoken about how having two Australian sides playing at once because of a cramped schedule degrades the value of international representation, while the Australian Cricketers Association has raised the argument that players miss out on bonus payments when series are lost.
This three-match series shifts to Geelong on Sunday.
Warner had been rested from the recent losing one-day series in New Zealand to prepare for the Indian tour.
The Sri Lankans have just come off a winning Twenty20 series in South Africa, and their confidence with the bat was obvious.
Cheered on by their strong local support, complete with beating drums and trumpets, Seekkuge Prasanna and Kapugedera shared in a 20-run stand off 15 balls to will their side to victory.
While skipper Upul Tharanga fell for a duck, the fearless Dilshan Munaweera and Niroshan Dickwella quickly put the Australians on the back foot with their aggressive stroke play in a brutal stand of 74 off 47 balls.
Lanky paceman Billy Stanlake, one of several Australians looking for a pay day in the Indian Premier League auction on Monday, began his first over with successive wides, and his second only got worse when he conceded 24 runs, including three boundaries off the first three balls by Munaweera and the indignity of being swept for six off the last by Dickwella.
Their 50-run partnership came off only 25 balls, and they flourished by using the pace of the deliveries off the fast bowlers. It was only when leg-spinner Adam Zampa was introduced that Australia were able to drag back the run rate - and make two key incisions.
Dickwella would fall when an attempted push through mid-wicket resulted in an edge carrying to gully, and Munaweera would exit soon after when his swivelled pull shot to the leggie was caught by Finch at mid-wicket.
The contest then tightened before a 60-run stand off 39 balls from Asela Gunaratne (52 off 37) and Milinda Siriwardana (15 off 11) again gave Sri Lanka the upper hand.
There was another twist when debutant allrounder Ashton Turner dismissed both batsmen in the 18th over with his off-spin but the tourists would hold on.
"There are a lot of positives. There are still two games left - we can win the series," Finch said.
Sent in to bat, Australia began safely when Finch and Michael Klinger, the latter making his international debut aged 36, put on 76 in 63 balls.
Klinger has been a stand-out opener in the short forms of the game for years, and was unlucky to not have stepped out for his country years ago. He would clip four boundaries in making 38 off 32 balls before an attempted swipe over mid-wicket off the left-arm wrist spin of Lakshan Sandakan - the hero of Sri Lanka's Test victory in Kandy over Australia last year - resulted only in a top edge and catch at gully.
Finch was in an aggressive mood and appeared to be setting himself to bat deep into the innings, having reached 43 off 34 balls. He had smacked the first ball of the tame medium pace of Gunaratne over wide long on for six and was set for a half-century, but fell two balls later when attempting a similar stroke, only this time to top edge a chance to short fine leg.
Australia's innings from that point lacked a burst of heavy hitting, something the injured Chris Lynn - one half of the Brisbane Heat's "Bash Brothers" - could have provided. Moises Henriques (17 off 10) looked to fill void but was caught at long on off having mistimed a half volley.
Travis Head, dropped by Tharanga on four, contributed a polished 31 off 25 balls but would be the first of successive wickets for paceman Lasith Malinga in the 19th over. The super slinger, in his first international Twenty20 for more than a year, would have Head snapped up at deep cover, and Turner caught at deep mid-wicket. The hosts would take 14 off the final over, with Sri Lanka left to rue some poor fielding.