The 35-year-old followed his 6-28 figures in the first innings with a further 6-54 in the second to finish with a match haul of 12-82, his 20th 10-wicket haul in Tests.
Muralitharan reached the landmark with the last wicket of the match as Syed Rasel spooned up a catch to Farveez Maharoof at mid off, sparking wild celebrations in the bowler's hometown stadium.
His final wicket dismissed Bangladesh for 176 and secured Sri Lanka a 3-0 whitewash with an innings and 193-run victory.
Muralitharan is chasing Australian Shane Warne's 708 wicket world record tally, a milestone he looks likely to pass during Sri Lanka's two test tour of Australia in November.
A veteran of 113 Tests, Muralitharan needed just 12 matches to go from 600 to 700 wickets.
The spinner's career has been plagued by controversy over the legality of his bowling action after being no-balled for throwing in Melbourne by umpire Darryl Hair in 1995.
Doubts still persist in some quarters, despite extensive biomechanical testing determining his action is legal according to current ICC regulations which permit a 15-degree elbow bend.