Sulieman Benn was unbeaten on 29 with the help of two sixes and Kemar Roach on 12 not out when play was called off due to rain after just 22.3 overs in the day.
The series is tied 0-0 following rain-hit Tests in Colombo and Galle. With just one day remaining, this Test is also heading for a draw.
The West Indies added 59 runs to their Thursday's total of 244-5, with left-arm spinner Herath being the only successful bowler. No play was possible on Friday due to rain.
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Herath has so far taken four wickets for 54 runs in the innings and spinner Ajantha Mendis two. Seamers Suranga Lakmal and Dilhara Fernando have bagged one wicket apiece.
Sri Lanka started the day with a double-spin attack and instantly got a wicket when Brendan Nash (67) was caught behind attempting to sweep Herath in the morning's second over.
Nash could add only five runs to his Thursday score of 62, hitting five fours in a 117-ball knock for his second half-century of the series.
Herath struck in his next ov