Eyewitnesses at the airshow, attended by about 4,000 people said the plane exploded when it hit the ground. Television footage showed flames and thick smoke.
Television reports quoted witnesses as saying the pilot might have tried to avoid hitting the crowds and instead chose to crash on a nearby field.
The ministry said the cause of the crash, some 240 kilometers (150 miles) east of Bucharest, was being investigated.
It named the dead pilot as Lieutenant-commander Florin Rotaru, 36.
Romania, a NATO member since 2004, received its first six F-16 fighter jets from Portugal in 2016, as it phases out its communist-era MiGs.
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