The twins, both males, were born at Pinnawala Elephant Orphanage in the central hills, an orphanage spokesperson said.
The mother, a 25-year-old elephant named Surangi, had given birth before in 2009 to a male calf. The father of the twins is a 17-year-old elephant who is also an orphanage resident.
Sri Lankan elephant experts said the previous birth of twins to a domesticated elephant was in 1941.
Pinnawala is one of Sri Lanka's biggest tourist attractions and was set up in 1975 to care for wild elephants that needed to be rescued and treated for various injuries.