The tourists, who won the first Test by an innings and 248 runs in Dhaka last week, bounced back from 49-2 to 196-2 at tea on the opening day at the Zahur Ahmed Chowdhury stadium.
Left-handed Sangakkara followed his 75 in the first Test with an unbeaten 96 that included 11 boundaries and two sixes.
Jayawardene, who scored 203 not out in Dhaka, continued his good form with 54 not out as the Bangladesh bowlers failed to take a wicket in the post-lunch session.
But they removed both openers in quick succession just after the drinks break to share the morning session which ended with Sri Lanka on 71-2.
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Openers Dimuth Karunaratne and Kaushal Silva managed just eight runs in the first eight overs and plodded to 30-0 after 15 when Bangladesh broke through.
Off-spinner Sohag Gazi, who shared the new ball with Al-Amin Hossain, claimed first Test century-maker Silva leg-before for 11.
Left-hander Karunaratne settled in to score 31 but then slashed at a wide ball from Al-Amin and was caught at point to make Sri Lanka 49-2.
Sri Lanka went into the Test without frontline left-arm spinner Rangana Herath and seamer Shaminda Eranga, who were both ordered to return home due to injuries.
They were replaced by off-spinner Ajantha Mendis and paceman Nuwan Pradeep, the only changes from the side that won the first Test.
Bangladesh brought in batsman Imrul Kayes, all-rounder Mahmudullah and left-arm spinner Abdur Razzak in place of Marshall Ayub, Rubel Hossain and Robiul Islam.