HELSINKI — Finnish police say they responded to the incident at Viertola school before 09:00 (06:00 GMT) on Tuesday and urged local residents to remain indoors.
A suspect, who police say was also aged 13, has been arrested.
The school has 800 students and 90 staff. Witnesses told public broadcaster YLE that two ambulances had left the scene.
In common with other Finnish schools, children had just returned to classes in Vantaa, north of the capital Helsinki, after the long Easter weekend.
The school has students aged seven to 15 of both primary and middle-school age on two separate sites.
As news emerged of the shooting parents gathered at the Jokiranta site where the incident happened.
Vantaa is Finland's fourth biggest city with some 240,000 residents.
Finland saw two deadly school shootings in a matter of months in 2007 and 2008, prompting the tightening of gun laws.
But it is widely known as a country of hunters and gun enthusiasts and has 430,000 licenced gun owners in a population of 5.5 million, according to government statistics.
In 2007, an 18-year-old student shot dead seven pupils and his head teacher in the town of Tuusula, then the following year another student shot dead nine pupils and a teacher in the town of Kauhajoki. — BBC