The passenger plane with 10 foreign passengers and two crew members went down on Sunday in Guanacaste, a province in the country's northwestern corner.
Local media reports suggested that those on board included five members of a single family.
There were no survivors.
Guillermo Solís said an investigation into the accident would start tomorrow. The cause of the crash is currently unknown.
Former Costa Rican President Laura Chinchilla wrote on Twitter that her cousin, Juan Manuel Retana, was among the dead.
"A death has interrupted our family celebration. Our beloved cousin died as part of the crew of the plane that crashed in Guanacaste. Courage to his children and siblings and may he rest in peace. You will stay in our hearts dear #Juanmanuel Retana" she said.