The hosts played better against South Africa, but will take the win after struggling for the first two sessions on the final day, leaving themselves six wickets to capture in the final session.
Man of the match Peter Siddle (4-45) led the attack with vigour, adding to his five wickets from the first innings.
The Victorian quick took the first four wickets of the day and snared his career-best match figures.
Mitchell Starc (5-63) was outstanding in the third session, reaching into the 140kmphs with precision yorkers and bouncers, and taking a well-deserved 4-17 in 38 balls to wrap up an insipid Sri Lankan tail.
Sri Lanka were bold on day five and were a chance to save the game at 4-186 at tea. It was déjà vu for the Aussies, who bowled 148 overs at South Africa in Adelaide after dominating the match and coming away with only a draw.
Once Siddle knocked over all-rounder Angelo Mathews (19) and Thilan Samaraweera (49) after the break, the visitors collapsed.
It was tough work for the quicks, who were one short after Tasmanian Ben Hilfenhaus succumbed to a side strain.
Skipper Michael Clarke soldiered on in the field despite suffering a hamstring strain while putting on a blistering half-century on day four.
Shane Watson was forced to bowl 27 overs in Hilfenhaus’s absence.
Before tea, Clarke was willing to try anything for a wicket, throwing the ball to wicket keeper Matthew Wade, who bowled his first ever over in first class cricket. And it was a maiden no less. He was deputised with the gloves by Phil Hughes.
Kumar Sangakkara (63 from 226 deliveries) was gallant before falling to Siddle.
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