The off-spinner claimed six wickets in each innings as the tourists, trailing by 369 runs, were shot out for 176 just before tea on the fourth day of the third and final Test here.
The ruthless Sri Lankans won all three Tests by innings margins to record their 10th successive victory over Bangladesh in as many matches.
Muralitharan signalled the win by having last man Syed Rasel caught in the deep for his 700th wicket, leaving him only nine scalps away from surpassing the retired Australian Warne's world record tally of 708.
The prolific 35-year-old could overtake Warne when Sri Lanka tour Australia for two Test matches in Brisbane and Hobart in November.
Muralitharan has now claimed 10 wickets or more in a Test match 20 times, giving him an astonishing 76 victims in nine Tests against Bangladesh.
It was Bangladesh's 43rd defeat in 49 matches since they gained Test status in 2000 with five draws and a lone win against fellow-strugglers Zimbabwe.
Bangladesh, who were 123-2 soon after lunch, lost their last eight wickets for 53 runs as they crashed to their seventh innnings defeat in 10 matches against Sri Lanka.
Left-hander Shahriar Nafees top-scored for Bangladesh with 64 but found little support at the other end when the Sri Lankans turned on the screws in the field.
The 21-year-old Nafees, a surprise omission from Bangladesh's one-day squad for the upcoming series against Sri Lanka, answered the selectors with nine boundaries in his third Test half-century.
Bangladesh began their second knock at the start of the day after Sri Lankan captain Mahela Jayawardene declared at the overnight score of 500-4, leaving Kumar Sangakkara unbeaten on 222.
Nafees and Javed Omar put on 47 for the first wicket, missing by one run their best opening stand in the series of 48 in the second Test.
Omar made 22 with the help of four boundaries when he edged fast bowler Lasith Malinga to Sangakkara at first slip.
Former captain Habibul Bashar helped Nafees put on 51 for the second wicket but could not break the poor streak that saw him aggregate just 54 runs in the previous five innings of the series.
Bashar, Bangladesh's most experienced Test batsman, made 15 when he stepped out to loft Muralitharan out of the ground, missed the line and was bowled.
Nafees added seven runs to his lunch score of 57 when he was beaten in flight by Muralitharan and edged a simple catch to Mahela Jayawardene in the slips.
Bangladesh captain Mohammad Ashraful, his team's main batting hope, plodded for an hour to score 19 when he played back to Muralitharan and was trapped leg-before.
Left-arm seamer Sujeewa De Silva removed Mushfiqur Rahim and Rajin Saleh in successive overs as Bangladesh slipped from 123-2 to 142-6.
Muralitharan then claimed three of the last four wickets as Sri Lanka won the rain-hit Test with more than a day to spare.