Dhananjaya de Silva batted steadily and along with captain Mathews, forged a Sri Lankan recovery with a stylish half-century.
Mitchell Starc dismissed debutant Avishka Fernando for a two-ball duck after Sri Lanka elected to bat first.
John Hastings dismissed half-centurion Dhananjaya de Silva to spark a Sri Lankan middle-order collapse.
Spinners Adam Zampa and Travis Head were amongst the wickets as Sri Lanka's middle order capitulated in the face of disciplined bowling.
Angelo Mathews survived a blow to his helmet off a Scott Boland delivery after the Australians had struck in quick succession to remove Kusal Mendis and Dinesh Chandimal.
Angelo Mathews retired hurt after suffering a calf strain midway through Sri Lanka's innings.
George Bailey scored an 85-ball unbeaten 90 to bat Sri Lanka out of the game.
David Warner became Pathirana's third scalp before Travis Head and George Bailey got together for a 100-run stand.
Australia chased down 213 with 19 overs to spare to clinch the five-match series with a game to spare.
Sachith Pathirana sent back Finch and Usman Khawaja in quick succession as Sri Lanka sought a comeback.
Aaron Finch laid the platform for Australia's chase with a bludgeoning 19-ball 55.
John Hastings ended with figures of 6 for 45 - his career best in ODIs - as Sri Lanka folded for a measly 212.