Earlier, Shaminda Eranga picked up four wickets and new-ball partner Suranga Lakmal claimed three. Mathews took the other wicket to fall to the pacemen.
Bangladesh lost four wickets in the morning before captain Mushfiqur Rahim (61) and all-rounder Shakib Al Hasan (55) stitched together an 86-run stand for the fifth wicket.
Left-hander Shakib's attacking innings came to an end when he was trapped leg before by left-arm spinner Rangana Herath who took two wickets.
Bangladesh also lost Nasir Hossain (four) in the next over as they were reduced to 150 for six.
Mushfiqur then combined with Sohag Gazi (42) in a 53-run partnership for the seventh wicket before the former was trapped leg before by Lakmal.
Bangladesh handed a test debut to 25-year-old opening batsman Shamsur Rahman who made 33.
Sri Lanka wicketkeeper Prasanna Jayawardene was a last-minute withdrawal due to the death of his father, Dinesh Chandimal taking the gloves for the visitors.
(Writing by Sudipto Ganguly, editing by Tony Jimenez)
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