"These attacks will not remain unanswered," the Azerbaijani Defence Ministry said in a statement on Thursday.
It added it was investigating the types of drones used in the attack, and "preparing the necessary response measures to protect the territorial integrity and sovereignty of the country and to ensure the safety of civilians and civilian infrastructure".
Azerbaijan and Iran already have tense relations over Baku's growing economic, energy and military ties to Turkey and Israel.
Iran, which has been retaliating for US and Israeli strikes since Saturday in an air war now engulfing the Middle East, also boasts a sizeable ethnically Azerbaijani population.
The Foreign Ministry earlier demanded in a statement that Iran "clarify the matter in the shortest possible time, provide an explanation and take the necessary urgent measures to prevent such incidents from recurring in the future".
"This attack on the territory of Azerbaijan contradicts the norms and principles of international law and contributes to increased tensions in the region," it said.
The Iranian ambassador to Azerbaijan was summoned to the ministry and received a formal note of protest.
Azerbaijan said one drone fell on the terminal building of the Nakhchivan International Airport, which is approximately 10 km across the border from Iran, and another drone landed close to a school building in a nearby village.
Video footage showed black smoke rising near the airport and damage to the skylight inside the terminal building.
The four injured were taken to hospital in stable condition, the Nakhchivan Health Ministry told Reuters.
Nakhchivan is a landlocked exclave of Azerbaijan bordering Armenia, Iran and Turkey.
NATO member Turkey and Azerbaijan signed an alliance agreement in 2021, which includes strengthening defence ties.
Turkey said on Wednesday that NATO air defences destroyed an Iranian ballistic missile headed into Turkish airspace.
