The hosts lost Michael Vandort and Kumar Sangakkara in the opening seven overs before reaching a 122-run target in the closing session, thanks to Malinda Warnapura (54 not out) and skipper Mahela Jayawardene (50 not out).
Sri Lanka had won the opening Test by an innings and 239 runs and India the second match by 170 runs.
India, trailing by 147 runs, were bowled out for 268 in their second innings despite gutsy half-centuries from a struggling Rahul Dravid (68) and an injured Venkatsai Laxman (61 not out).
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The tourists were without paceman Ishant Sharma, who batted with a runner but was unfit to bowl after falling on his follow-through on Saturday.
Off-spinner Harbhajan Singh shared the new ball with Zaheer Khan and struck with his fourth delivery when he bowled out-of-form opener Vandort (eight), who finished the series with just 39 runs.
Zaheer claimed a big wicket in his fourth over when he removed Sangakkara, who drove straight to Gautam Gambhir at mid-off.
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