Just 29 overs were bowled during the day in which Bangladesh, sent in to bat for the third successive time in the series, groped to 72-4 in dark, gloomy conditions at the Asgiriya stadium.
Off-spinner Muttiah Muralitharan took 2-10 in nine overs, taking his career tally to 690 wickets and just 18 Saway from equalling retired Australian spinner Shane Warne's world record haul of 708.
Muralitharan, 35, aims to overtake Warne when Sri Lanka tour Australia for two Test matches in November.
The seasonal monsoon rains had not interfered with the first two Tests in Colombo, allowing Sri Lanka to win both matches by innings margins to take a decisive 2-0 lead in the series.
But inclement weather caught up with the cricket in the popular hill resort of Kandy as only 12 overs were bowled in the morning session, 5.4 overs in the post-lunch period and 11.2 overs after tea.
That, however, was enough for the Sri Lankan bowlers to run through the Bangladeshi top order with the wickets of Javed Omar, Habibul Bashar, Rajin Saleh and Shahriar Nafees.
When bad light ended play, captain Mohammad Ashraful was on four and Tushar Imran was on six.
Sri Lankan captain Mahela Jayawardene won his third consecutive toss in the series and did not hesitate to ask Bangladesh to take first strike once again.
The tourists were dismissed for 89 on the opening day of the first Test and were then skittled out for their lowest total of 62 at the start of the second Test.
But Bangladesh fared better in overcast conditions despite losing Omar in the third over of the morning, trapped leg-before by fast bowler Lasith Malinga for eight.
Nafees and Bashar put on 40 for the second wicket between the rain breaks when seamer Farveez Maharoof forced Bashar to glove a catch to wicket-keeper Prasanna Jayawardene after lunch.
When play resumed after tea, Muralitharan struck twice in quick succession to reduce Bangladesh to 64-4.
Rajin Saleh, who faced 29 balls without scoring, edged a straight ball to Mahela Jayawardene in the slips.
Left-handed Nafees looked comfortable while making 29 when he hit out against the wily Muralitharan and offered a tame catch to Sujeewa De Silva at mid-off.
Sri Lanka made three changes from the side that clinched the series in the second Test at the P. Sara stadium in Colombo last week.
The hosts recalled opener Upul Tharanga and new-ball bowlers Maharoof and De Silva.
The trio replaced Chaminda Vaas, who has returned to his English county Middlesex, opener Malinda Warnapura and fast bowler Dilhara Fernando.
Bangladesh brought in middle-order batsman Tushar Imran and left-arm seamer Syed Rasel in place of Mehrab Hossain and Mohammad Sharif.