The mid-morning blasts hit Beirut's upscale neighbourhood of Janah, a stronghold of the Iranian-backed Shiite Muslim Hezbollah group. One explosion blew out the large black main gate of the Iranian mission, damaging the three-story facility.
It was not immediately clear if the attack was related to the conflict next door in Syria, but attacks have targeted Hezbollah strongholds in recent months in what many see as retaliation by Sunni extremists for the militant Shiite group's role in the Syrian civil war.
Iranian Ambassador Ghazanfar Roknabadi identified the dead diplomat as Sheikh Ibrahim Ansari. Speaking to Hezbollah's Al-Manar TV, he said Ansari took his post in Lebanon a month ago and was overseeing all regional cultural activities.
Lebanese Health Minister Ali Hassan Khalil said 23 people were killed and 146 wounded in the blasts.
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Debris was scattered on the street and cars were on fire as people ran away from the chaotic scene. The second blast was meters (yards) away from the embassy building.
An armed guard at the embassy told AP that the first blast was believed to have been carried out by a suicide attacker who rode a motorcycle and blew himself up outside the gate.
The guard, speaking on condition of anonymity because he was not authorised to talk to media, said the other explosion, which caused much more damage, was likely a car bomb.
Lebanese security officials confirmed the two bombings were both suicide attacks. They spoke on condition of anonymity in line with regulations.