Zimbabwe, whose spinners had given them a chance on a lifeless pitch at least to try to prevent the whitewash, replied with 133, hardly the stuff of hot batsmanship and indeed they lost their last four wickets for just three runs.
This narrow triumph followed two previous close calls, where Sri Lanka had won by two wickets and then by five runs.
The first two matches in the series had ended quickly for the Sri Lankans as they bundled out the home side for 67 and 124.
Zimbabwe almost snatched it, however, in Sunday's final effort before their late collapse.
With Prosper Utseya and Keith Dabengwa putting on 72 for the seventh wicket to reach 130, Sri Lanka were showing signs of anxiety until Muralitharan captured the tail-enders amid much celebration from the visitors.
Sri Lanka's innings was a modest effort and neither team managed to achieve a score of 200 runs in the entire series.
Captain Mahela Jayawardene had a miserable time here, scoring just 14 runs in four innings.