Lt Gen Hussein Ishaq died yesterday as rebels attacked a Syrian air defence base near the town of Mleiha, the official said.
He spoke on condition of anonymity because he wasn't authorised to brief journalists about Ishaq's death.
The Britain-based Syrian Observatory for Human Rights also confirmed Ishaq's death. It said he died from wounds he suffered during Saturday's fighting.
The Observatory, which bases its reports on a network of activists on the ground, said Ishaq was killed in clashes with fighters from the Nusra Front, the al-Qaida affiliate in Syria, and other Islamic rebel groups.
Ishaq is one of a few high-ranking military officers to be killed in the Syrian war. It's unclear if his death will have any wider strategic or military impact on the war itself, however.