Mahela Jayawardene hit a classy 105 and Angelo Mathews made 93 to hold off Australia, but the rest of the batting caved in.
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It was Michael Clarke's first Test win as Australian captain, having lost to England by an innings in Sydney in January when he stood in for the injured Ricky Ponting.
The emphatic win in Galle helped Ponting set a new benchmark of becoming the first cricketer to have been part of 100 Test victories.
The 36-year-old, the team's frontline batsman, was already Test cricket's most successful captain with 48 wins when he stepped down in April to make way for Clarke.
Jayawardene and Mathews, who came together on Friday afternoon with Sri Lanka tottering on 68-5, put on 142 for the sixth wicket in a courageous fightback.
The pair batted through the entire morning session, defying the A