The stylish left-hander added 178 for the third wicket with Mahela Jayawardene (72) after Bangladesh dismissed openers Dimuth Karunaratne (31) and Kaushal Silva (11) cheaply.
The 36-year-old Sangakkara, who went to his half-century with a six off spinner Mahmudullah, brought up his hundred with a boundary to fine leg off the same bowler.
Jayawardene also looked set for his 34th test hundred but was out leg before to Mahmudullah after playing across the line.
Sangakkara and Dinesh Chandimal (27) then put 67 for the fourth wicket before the latter was caught at point as he danced down the wicket to left-arm spinner Shakib.
All-rounder Shakib then bowled captain Mathews for five to peg back Sri Lanka late in the day.
Bangladesh, crushed in the first test in Dhaka by an innings and 248 runs, dropped Marshall Ayub, Rubel Hossain and Robiul Islam and brought in batsman Imrul Kayes, all-rounder Mahmudullah and left-arm spinner Abdur Razzak.
Paceman Nuwan Pradeep and spinner Ajantha Mendis replaced the injured Shaminda Eranga and Rangana Herath in the Sri Lankan team.
(Writing by Sudipto Ganguly, editing by Tony Jimenez)
