Zimbabwe were left teetering on 180 for seven in their second innings, 310 runs behind the tourists, who need three wickets on the final day at the Harare Sports Club to wrap up a series sweep.
Herath, who claimed 10 wickets in a test match for the seventh time, exploited a turning pitch after Zimbabwe were put into bat just before tea, chasing 491 runs for victory. Only Craig Ervine, who was 65 not out at the close of play, offered any resistance.
Sri Lanka started the fourth day on 102 for four with opener Dimuth Karunaratne falling 12 runs short of a century as he was trapped leg before wicket by Chris Mpofu.
They declared on 258 for nine when Kusal Perera was dismissed for 62. Zimbabwe captain Graeme Cremer was again the pick of the home bowlers with figures of 4-91.
(Reporting by Mark Gleeson in Cape Town; Editing by Toby Davis)
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