The tourists, who resumed at their overnight score of 72-4, lost their last six wickets for 33 runs as off-spinner Muralitharan and seamer Sujeewa De Silva ran through the lower order.
Heavy rain soon after Bangladesh's innings ended delayed Sri Lanka's reply, further disrupting a Test in which only 29 overs were bowled on the first day.
Sri Lanka are seeking a 3-0 sweep of the ongoing series after winning the first two Tests in Colombo by innings margins.
Muralitharan's 59th haul of five wickets or more in an innings took his career tally to 694, leaving him just 14 away from equalling Australian Shane Warne's world record of 708 wickets.
Muralitharan, who still has one more innings against Bangladesh in this match to increase his tally, hopes to overtake the retired Warne when Sri Lanka play two Tests in Australia in November.
When play began 30 minutes late due to a wet outfield caused by overnight rain, Bangladesh added 26 runs when De Silva forced Tushar Imran to edge a