Tillakaratne Dilshan's superb 125 helped Sri Lanka post an imposing 302 for nine in Pallekele, Kusal Perera adding 56 and Kumar Sangakkara 48 as Abdur Razzak responded with a five-wicket haul.
Bangladesh started well before rain intervened and though some rash slogging on the resumption put their position in doubt, Nasir (33 not out) saw them home with an over to spare.
Sri Lanka won the toss and Dilshan and Perera quickly piled on the runs, with 50 already on the scoreboard in the seventh over. The 100-run partnership arrived 10 overs later with half-centuries following for both batsmen soon after. The stand was broken by Mahmudullah, whose delivery was misread by Perera and pushed to Nasir at point.
Dilshan formed a solid partnership alongside Sangakkara, taking the score to 203 when the latter became Razzak's first victim. Razzak struck again to remove Thisara Perera for four but Dilshan continued, bringing up his century from 110 balls. At the other end, Angelo Mathews (six), Jeevan Mendis (five) and Dinesh Chandimal (nought) fell cheaply.
Shahadat Hossain took the prized wicket of Dilshan, caught by Nasir this time at mid-wicket, and Razzak removed Nuwan Kulasekara and Sachithra Senanayake to finish with five for 62.
In the Tigers' reply, an expensive third over featured two Anamul Haque boundaries and a set of five wides. Mohammad Ashraful hit off-side boundaries off Mathews and Lasith Malinga and the 50 partnership came up from the first ball of the eighth over. Ashraful was dismissed for 29, lbw to Senanayake to end a stand of 75.
Rain set in less than an over later and when it relented, following a delay of more than two hours, Bangladesh returned to chase 183 from 27 overs, needing a further 105 runs from 80 balls. They took 11 from the first four thanks to two fours for Jahurul Islam but Anamul was dismissed for 40 when he skied Mathews to cover.
The reckless approach continued as Jahurul threw away his wicket with a slash to wicketkeeper Sangakkara after 12 runs had already come from Mathews' over. Mushfiqur fell to a kamikaze run-out and Mahmudullah was lbw after missing a reverse sweep at Senanayake.
Nasir showed the way with some more measured aggression but Mominul Haque and Ziaur Rahman fell in the same Malinga over to leave their team 166 for seven. The penultimate over began with two Nasir boundaries and a wide and after Nasir's sixth four helped him to an unbeaten 33, Sohag Gazi top-edged for the winning boundary.
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- Tillakaratne Dilshan
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