Bangladesh were bowled out for 176 shortly before tea on the fourth day after Sri Lanka had declared on their overnight total of 500 for four, condemning the tourists to a 3-0 series whitewash following a pair of equally heavy defeats in Colombo.
Cheered on by a raucous hometown crowd, Muralitharan followed his six for 28 figures in the first innings with six for 54 in the second to capture his 20th 10-wicket haul in Tests.
Muralitharan reached the landmark when he took the last wicket of match, tricking Syed Rasel into spooning up a catch to Farveez Maharoof at mid off. The 35-year-old is now just eight wickets behind Australian leg spinner Shane Warne's world record tally, a figure he appears set to pass during Sri Lanka's two-Test tour of Australia in November.
Muralitharan was later named man of the match and the series having claimed 26 wickets at an average of 10.84.
Bangladesh provided some resistance early on before pace bowler Lasith Malinga ended a 47-run opening partnership, removing Javed Omar for 22 when Kumar Sangakkara held on to a sharp slip catch.
Muralitharan's first wicket of the innings came shortly before lunch as former captain Habibul Bashar slogged across the line and was bowled, ending a 51-run stand for the second wicket.
After lunch, Muralitharan ended opener Shahriar Nafees's 115-ball stay at the crease when the left-hander edged a sharp off break to skipper Mahela Jayawardene at slip for 64.
Muralitharan also dismissed Bangladesh captain Mohammad Ashraful, who was pinned lbw for 19.
Left-armer Sujeewa de Silva, returning to the attack after an accurate post-lunch spell from Farveez Maharoof, had Mushfiqur Rahim (1) caught at point and Tushar Imran (17) snapped up at slip.
With the Bangladesh top order gone, the lower order folded quickly with the last eight wickets adding just 53 runs. Bangladesh's defeat soon became inevitable and the only question was whether Muralitharan would reach another milestone in front of a cheering home crowd.
He scripted the end perfectly though, clean bowling Shahadat Hossain and then deceiving last-man Rasel to finish off the game.

SCOREBOARD
BANGLADESH (I Innings): 131
SRI LANKA (I Innings): 500/4 decl.
BANGLADESH (II Innings):
Omar c Sangakkara b Malinga 22
Nafees c M J’dene b M’tharan 64
Bashar b Muralitharan 15
Ashraful lbw Muralitharan 19
Mushfiqur c Tharanga b De Silva 1
Tushar c M J’dene b Malinga 17
Saleh c M J’dene b De Silva 0
Mortaza c sub b Muralitharan 8
Rafique (not out) 0
S Hossain b Muralitharan 5
Rasel c Maharoof b M’tharan 4
Extras (B-4, LB-1, NB-13, W-3) 21
Total (all out, 59 overs) 176
Fall of wickets: 1-47, 2-98, 3-123, 4-138, 5-138, 6-142, 7-166, 8-167, 9-172.
Bowling: Lasith Malinga 10-0-46-2, De Silva 12-4-34-2, Maharoof 16-7-37-0, Mutaiah Muralitharan 21-5-54-6.