It is unclear how the mishap occurred, Government Railway Police (GRP) said, adding they are investigating whether the trio were crossing the railway tracks at the time of the accident.
The deceased are identified as Maltidevi (25), Nancy (4) and Sanajan (one-and-half-year-old).
Their mutilated bodies are sent to government hospital for postmortem, they said.
The deceased were residents of Maneri village in Titwala east.
A case of accidental death has been registered by police.