Just 54 minutes of play was possible before lunch on the first day at the Asgiriya stadium in the central Sri Lankan city of Kandy. Bangladesh, sent in to bat in overcast conditions, made 48-1 in 12 overs.
Shahriar Nafees was unbeaten on 16 and Habibul Bashar was on 18 after Javed Omar had fallen leg-before to Lasith Malinga for eight.
Sri Lanka are chasing a 3-0 series sweep after winning the previous two Tests in Colombo by innings margins.
Sri Lankan captain Mahela Jayawardene won his third successive toss in the series and did not hesitate to ask Bangladesh to take the 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's batsmen refused to surrender once more as Nafees and Bashar put on 38 for the unbroken second wicket.
Left-handed Nafees has so far hit two boundaries and Bashar, battling to strike form, has three fours to his credit.
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 Farveez Maharoof and Sujeewa de Silva on a seamer-friendly wicket.
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.