A 12-year-old boy and a 15-year-old girl died at the Colombo National hospital where 10 others were being treated after being rescued, hospital spokeswoman Pushpa Soysa told AFP.
Police said hundreds of troops had joined the search for survivors after the disaster at Kolonnawa on the northeastern edge of the capital.
President Maithripala Sirisena ordered troops and police to join firefighters in the rescue after the 300-foot (91-metre) high dump caught fire and collapsed, officials said.
Police said the true scale of the damage remained unclear.
"A search for survivors is under way," the police said in a statement.
Dozens of homes collapsed after heavy rains overnight caused the garbage mountain to shift, officials said. It became further destabilised after a fire broke out, triggering landslides that buried dwellings.
Military spokesman Roshan Seneviratne said 100 soldiers were already digging through mounds of trash. Heavy earth moving equipment was also being deployed, he added.
"We think about 40 homes have been destroyed," a disaster management official told reporters.
Roughly 800 tonnes of solid waste is added daily to the open dump, angering residents who live nearby.
Sri Lanka's parliament was warned recently that the 23 million tonnes of garbage rotting at Kolonnawa was a serious health hazard.
Efforts are underway to build an electricity plant that could transform the solid waste into fuel.
Friday's fire broke out as the country marked its traditional Sinhala and Tamil New Year and most people were in their homes celebrating.
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