This is the first series loss for Sri Lanka at home since 2005.
Clarke made 112 as the tourists, trailing by 157 runs after the first innings, were all out for 488 in their second knock on the fifth day, a lead of 331 runs.
Michael Hussey followed two successive hundreds in the series with 93 after helping Clarke add 176 runs for the fifth wicket to frustrate Sri Lanka's bid for a series-levelling win.
The second Test in Pallekele ended in a rain-affected draw.
Clarke, who went 23 innings without a Test century since making 168 against New Zealand in March last year, hammered 14 boundaries and three sixes to end his drought.
The hundred left him on the doorstep of a series win in his first Test tour as captain after replacing Ricky Ponting at the end of the World Cup in April.
Clarke eventually lost concentration following a brief verbal skirmish with Sri Lanka's Kumar Sangakkara and holed out in the deep to give left-arm spinner Rangana Herath his fifth wicket of the innings.
Left-hander Hussey continued his remarkable form, taking his run aggregate past 450 runs in five innings with two centuries.