Authorities in Arish, the capital of North Sinai province, were on the hunt for gunmen who fired at the patrol car, state news paper Al-Ahram reported on Wednesday.
Attacks on security forces have been frequent in Egypt since the army, led by general-turned-President Abdel Fattah al-Sisi, ousted Muslim Brotherhood President Mohamed Mursi in 2013 after mass protests against his rule.
The violence has been concentrated in the Sinai Peninsula, where Egypt is fighting an insurgency, but has also expanded to hit Egypt's Coptic Christians, the country's largest minority.