Earlier Bangladesh, who started the day on 56-4 in response to Sri Lanka s 222, faltered against Dananjaya s mystery spin that rattled the hosts middle and lower-order.
Silva finished on 58 off 94 balls, which included nine boundaries, while Lakmal remained undefeated on seven off 22 deliveries, which included a solitary boundary.
Bangladesh's highest successful fourth-innings chase was 215 in 2009 against the West Indies.
More likely, Sri Lanka looks set to record a 16th win in 20 tests against Bangladesh, as early as Saturday.
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The first Test in Chittagong ended in a draw with Mominul Hoque scoring tons in both innings.
Dhananjaya de Silva (28) made a decision to attack the Bangladesh bowlers with three boundaries in the 10th over as Sri Lanka crossed 50 runs in the 15th over. "I don't know why they gave a wicket like this", he said. It is common sense but cricket is a amusing game.
"We bowled them out for 100 and we wanted to get more than 300-plus runs actually. We passed it. Tomorrow morning we will add as much runs as possible to the target".
Every run Sri Lanka could get would be valuable with the pitch showing no signs of let up for the batsman. "On Sri Lanka's soil it is always tough to chase something like 200, so when we can chase nearly 200 in Sri Lanka why can't we chase 300-plus in our own backyard?"
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He hit nine fours in his nearly chance-less knock, facing 94 balls as his teammates struggled in the middle.
Sri Lankan wicket-keeper Niroshan Dickwella had a moment to forget during their Test match against Bangladesh as he dropped one can only be described as a clanger.
Dimuth Karunaratne and Dhananjaya de Silva (28) played contrasting knocks with the latter moving the game forward while the former showed dogged resistance. Offspinner Mehidy Hasan, on the other hand, kept posing questions and bowled a tight spell but runs came at a decent rate from the other end, allowing Sri Lanka to extend their lead.
Roshen Silva and captain Dinesh Chandimal stabilised the innings with a 51-run partnership before Chandimal was given out lbw off the bowling of Mehidy for 30.
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