The car of the second field army chief "came under heavy fire, which led to clashes between security forces and the terrorist elements," the spokesman said in a statement, without identifying the commander.
Militants attacked the commander's convoy while he was checking Shaykh Zuwayed area near the border with the Gaza Strip.
One girl was hurt during the exchange of fire and died later in hospital, the army said.
"There have been several attacks in Sinai on army and police officers to create chaos in the area and harming the stability and national security of Egypt," the army said.
Leading Muslim Brotherhood leader Mohamed al Beltagy said that violence in Sinai will stop as soon as Morsi is reinstated as Egypt's president.