Herath followed up his five-wicket haul in the first innings with 5 for 45 in the second as Zimbabwe, set 491 to win, went to stumps on the fourth evening on 180 for seven.
On day three: Five-star Herath among the elite
The feat not only took Herath to 50 wickets in the calendar year -- in just his eighth Test -- but also gave him the seventh 10-wicket match haul in his career.
The tourists began the fourth day on 102 for four in their second innings, with an overll lead of 334, and spent the first half of the day extending that advantage.
With Dimuth Karunaratne anchoring the innings with a patient 88 and Kusal Perera adding a quickfire 62 towards the end, Sri Lanka were able to declare their second innings on 258 for nine to set Zimbabwe 491 to win.
De Silva puts Sri Lanka in charge
Left with 136 overs to bat in order to save the match, Zimbabwe were soon reduced to 39 for three before a 74-run stand for the fourth wicket between Sean Williams and Craig Ervine briefly revived them.
Once Williams edged to slip off Lahiru Kumara to fall for 45, Sri Lanka were able to chip away at the middle order and move within three wickets of a 2-0 series sweep.
Ervine reached stumps unbeaten on 65, with Donald Tiripano for company.
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