Thirimanne (155 not out) and Chandimal (116no) added their own three figure scores to build on Kumar Sangakkara's hundred as the hosts declared on an imposing 570 for four.
Bangladesh looked to be in trouble when they slipped to 65 for two in reply but a fluent 65 not out from Mohammad Ashraful in his first Test since 2011 guided the Tigers to 135 without further loss at the close.
Mushfiqur Rahim's men still have some way to go before they are out of the woods but Ashraful has given them a decent start. It has been a forgettable time for their bowlers although they were badly let down by their fielders, with Chandimal dropped three times before advancing beyond 12.
They at least enjoyed a decent start to the day, as Sri Lanka captain Angelo Mathews departed for 27 after returning a catch to Abul Hasan. Chandimal was dropped in the slips off Hasan when on five and then again at fine leg off Shahadat Hossain when on nine.
It seemed he could not give his wicket away as he then watched Jahurul Islam put down a more difficult chance off Sohag Gazi when on 12. At this point, Chandimal perhaps understood he had an air of invincibility about him as he quickly began to open up. Where his first 12 runs required 45 balls, he made 43 off the next 48 to reach lunch on 55.
While all that was going on at one end, Thirimanne brought up his ton with a lofted shot over mid-on, reaching the mark in 193 deliveries with nine boundaries. Chandimal's century came off 136 balls with 13 boundaries, including two maximums, but he was dropped three overs later by Mahmudullah off Gazi at long-on.
By then it hardly mattered and 40 minutes before the scheduled tea break, Sri Lanka declared with a daunting task in front of the tourists. Jahurul Islam got the Tigers off to an ideal start with a four through midwicket off the first ball of their reply and added another two boundaries in as many balls shortly afterwards.
He was the first wicket to fall, however, when he gloved Shaminda Eranga behind for 20. Ashraful settled into his stride after tea with a four through the covers and continued to take the attack to Sri Lanka's bowlers.
Debutant Anamul Haque was rather more measured at the other end and had only made 13 off 68 balls by the time he was bowled by Ajantha Mendis. Ashraful struck back-to-back boundaries in the spinner's next over and reached his half-century with a four through midwicket. Mominul Haque provided decent support and he will resume on 35 not out.
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- Mohammad Ashraful