A spokesman for Spain's defence ministry said the pilot, Ladislao Tejedor Romero, 35, an experienced jet pilot and assistant to Defence Minister Pedro Morenes, died of his injuries in the serious burns unit of Getafe hospital.
Some 3,000 people were at Cuatro Vientos airfield watching a showcase of aerial acrobatics and vintage aircraft when the plane, one of the first jet-propelled planes to be manufactured in Spain, crashed, sending a fireball and thick black smoke into the air.
Florenza said the cause of the crash was not known.
State broadcaster RTVE showed images of a large plume of smoke rising above hangars and interviewed witnesses who described the plane diving and crashing into buildings.
Spain's airport authority AENA said no one else other than the pilot was injured.
Europa Press news agency, however, reported quoting emergency services that two people, a rescue worker suffering from burns and a 57-year-old man suffering from an anxiety attack, believed to be the pilot's father, were transferred to hospital. A third rescue worker suffering from smoke inhalation received first aid at the scene, the agency said.