The aircraft, a single-propellor Cessna 208 Caravan belonging to the Nature Air domestic airline, came down in a mountainous area near the Pacific coastal beach town of Punta Islita in the country's Guanacaste peninsula, the spokesman, Carlos Hidalgo, said on his Facebook page.
"It is a private plane with 10 foreign passengers and two local crew members," a separate security ministry statement said.
Hidalgo published images of the crash site, showing flaming wreckage strewn across wooded terrain.
All the bodies were burned, Hidalgo told national television station Channel 7.
"I have the deaths of the 12 occupants confirmed," the head of Costa Rica's civil aviation agency, Enio Cubillo, told La Nacion newspaper.
The daily gave a list of passenger names, including five who shared the same last name, suggesting they were all related.
It reported that the plane had apparently taken off from a small sealed airstrip in Punta Islita at 10:30 am (1630 GMT) and crashed shortly afterwards for reasons yet to be determined.
It had been scheduled to land at San Jose's main Juan Santamaria airport at 10:55 am (1655 GMT).
The US embassy said it was aware of the crash and was "trying to confirm if any US citizens are on board."
Americans are by far the biggest nationality to visit Costa Rica, a popular tourist destination, especially in December and January when they flee freezing winter conditions at home for tropical climates.
Guanacaste is a magnet for many visitors to the Central American country, featuring pristine beaches and nearby jungle.
Nature Air is one of the small domestic airlines serving the area, typically flying tourists in a hurry to and from the capital San Jose.
This time of year is particularly busy in Costa Rica's coastal resort areas as tourists and locals alike spend end-of-year holidays in the sun.
The Cessna 208 turboprop plane is made in the United States.