Dilshan, who top-scored with 162 in the first innings, hit 81 while Samarweera scored 72 as the visitors reached 296-4 in their second innings at close for a massive overall lead of 472 runs.
Kumar Sangakkara also played a vital knock of 54 to tighten the screws on the hosts, who had scored 208 in their first essay in reply to Sri Lanka's 384.
The day's final session belonged clearly to Sri Lanka with their batsmen adding 109 runs without losing any wicket.
Dilshan, dropped on 55 off Mashrafe Mortaza, hit nine fours in his 114-ball knock and dominated the 131-run unbroken stand for the fifth wicket with Samarweera.
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Sangakkara initially paved the way for the middle-order with his sensible 103-ball effort containing four fours.
He reached his 28th Test half-century with a pulled four to the midwicket before playing on to his stumps off part-time spinner and captain Mohammad Ashraful (2-29).
Sangakkara also shared a 68-run stand for the third wicket with skipper Mahe