The off-spinner grabbed 4-30 in 23 overs as England, replying to Sri Lanka's modest 188, slumped to 186-6 by tea before heavy rain washed out the rest of the day's play.
The weather gods in Muralitharan's home town of Kandy held up his bid to become Test cricket's most successful bowler, but the 35-year-old is unlikely to be denied the honour when play resumes on Monday.
Muralitharan looked unplayable on the uneven pitch at the Asgiriya stadium as the batsmen groped to negotiate the alarming turn and bounce he extracted from the surface.
Paul Collingwood, unbeaten on 14, was England's last remaining hope to gain a substantial lead on a wicket where 16 wickets have already fallen on the first two days.
Ryan Sidebottom was the other batsman at the crease, on one.
Muralitharan, who went into the match needing five wickets to overtake Warne, had debutant Ravi Bopara caught behind by wicket-keeper Prasanna Jayawardene in the post-lunch session to claim his 708th victim.
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Sri Lank