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The feisty opener hit a boundary for the winning runs and remained unbeaten on 102 for his first ODI ton. Tillakaratne Dilshan plundered 51.
The tourists, who suffered a 107-run defeat in Melbourne on Friday in the first of the five-match ODI series, reached the target of 171 with 59 balls to spare.
Video highlights: Australia v Sri Lanka, second ODI
Australia had Upal Tharanga caught behind off Clint McKay in the third ball of Sri Lanka's innings with no runs on the board.
The hosts mixed up spin with speed but failed to make another breakthrough until Glen Maxwell caught Dilshan off Ben Cutting with the score already on 137.
Debutant Kushal Perera, who held four catches behind the wickets, clobbered 14 quick runs with Thirimanne in cruise control.
Sri Lanka's seamers led by slinger Lasith Malinga had skittled out Australia setting the stage for an ultimately easy victory.
Mahela Jayawardene won an important toss after heavy rain delayed the start and notched his first triumph on the tour after a 3-0 whitewash in the Tests.
Veteran wicket-keeper Brad Haddin top-scored for Australia with 50 while Malinga claimed three wickets for 32.
Sri Lanka effectively shackled Australia from the outset and picked up regular wickets. Opener Aaron Finch was first to go, caught off Angelo Mathews for just four.
Nuwan Kulasekera then had opening match centurion Phillip Hughes plumb lbw for three and Australia were in trouble on two for 12.
Skipper George Bailey struggled to 26 before Thirimanne took a leaping catch off a Malinga delivery.
David Hussey looked the man for a tough job, picking up occasional boundaries and putting on 39 with Bailey.
But Hussey only got to 29 when he hesitated going for a single and Dilshan knocked off the bails from mid-wicket.
Australia's inexperienced side needed a big innings from Haddin but he struggled as partners disappeared.
Steve Smith played a couple of good shots but went for eight, edging a Thisara Perera leg cutter to the keeper. Maxwell never looked set and nicked a fine ball from Mathews to the wicketkeeper.
Cutting, another debutant, brought some composure and quickly hit his first international boundary.
His seventh-wicket partnership with Haddin for 57 proved Australia's best, but another edge to the keeper ended his innings on 27.
Kane Richardson on his debut and on his home ground was lbw for a duck off a deceptive slower ball from Malinga.
Clint McKay fended off the hat-trick ball but was given out for four after a review of a fine outside edge caught behind the wicket again by Kushal Perera.
Haddin tried to attack and struck a big six off Thisara Perera but on 50 he holed out to Thirimanne at midwicket off the bowling of Ajantha Mendis to close the innings.
The third game in the series takes place in Brisbane on Friday.
