Bangladesh could not start their innings due to rain after bowling out Sri Lanka for 311 runs in 49.5 overs, thanks to a hat-trick of fast bowler Taskin Ahmed, who finished with 4-47.
The wash-out means Bangladesh now cannot lose the series after winning the first match by 90 runs on Saturday.
Needing a win to stay alive in the tournament, Kusal Mendis laid the foundation for a strong total for Sri Lanka with 102 off 107 balls before Taskin's late burst halted their progress at the Rangiri International Stadium.
Mashrafe Mortaza took opener Danushka Gunathilaka early for nine but Mendis shared 111 runs with skipper Upul Tharanga for the second wicket to put Sri Lanka in a strong position.
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A direct hit from Mahmudullah helped Bangladesh separate the duo as Tharanga was run out for 65, trying to complete a single of a no-ball off Mustafizur Rahman.
Mendis struck a six and a four in the next two balls to complete his fifty before racing to his century in 102 balls with a single off Mortaza.
Milinda Siriwardana also made a useful contribution with 30 runs.
Sri Lanka made three changes to their squad, bringing in Nuwan Kulasekara, Nuwan Pradeep and Dilruwan Perera for Sachith Pathirana, Lahiru Kumara and Lakshan Sandakan.
The third and final match of the series will be held in Colombo on April 1.
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