Sri Lanka, trailing by 68 runs, were 1-116 in their second innings at stumps on the penultimate day of the opening Test against Bangladesh in Galle on Monday.
Tillakaratne Dilshan was unbeaten on 63 and Kumar Sangakkara 49 not out.
Bangladesh were earlier bowled out for 638 in their first innings in reply to Sri Lanka’s 4-570 declared.
Skipper Mushfiqur Rahim top-scored with 200, the first Test double-century by a Bangladeshi batsman.
Mohammad Ashraful (190) and Nasir Hossain (100) were the other main scorers.
Rahim’s memorable moment came in the second over after lunch when he pushed paceman Nuwan Kulasekara for a single to the covers on a day when Mohammad Ashraful (190) missed out on his own Test double-hundred.
Wicketkeeper-batsman Rahim, 24, was trapped leg-before by Kulasekara in the same over, but not before helping Bangladesh surpass their previous highest Test total of 556 against the West Indies in Dhaka last year.
Bangladesh, who dominated the match for a second successive day with their solid batting, also became only the third team after India and Pakistan to score 600 or more in a Test innings in Sri Lanka.
Rahim, 152 overnight, hit one six and 22 fours in his brilliant 321-ball knock.
He put on a record 267 for the fifth wicket with Ashraful and 106 for the next with Hossain.
Bangladesh’s previous highest stand for any wicket in Tests was 200 between Tamim Iqbal and Junaid Siddique against India in Dhaka in 2010.
Hossain was caught at mid-wicket by Sangakkara off part-time spinner Dilshan after hitting nine fours.