Iranian media reports said Colonel Reza Zarei was killed along with two fighters from Lebanon's Iran-backed Hezbollah group.
The strikes come just days after alleged Israeli airstrikes targeted sites in Homs in central Syria.
Syrian air defenses shot down a number of missiles, state media said, without giving details on the targets hit.
Syrian state media said only material damage was caused by the airstrikes.
The attacks on an IRGC operational headquarters destroyed the building, local media reported.
Syrian air defenses were activated in the Damascus area during the strikes.
Three security sources and Iranian state media said an Israeli air strike outside Damascus on Monday killed Sayyed Razi Mousavi.
Iranian state-owned media confirmed the death of IRGC commander Seyed Razi Mousavi, identified as "a senior advisor" in Syria.
Jordan strikes Iran-backed drug smugglers in Syria as part of counter-narcotics operations.
Ambulances reportedly rushed to the area shortly after the strikes.
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