The mid-morning blasts hit Beirut's upscale neighbourhood of Janah, a Hezbollah stronghold. One explosion blew out the large black main gate of the Iranian mission, damaging the three-story facility. It was not known if anyone inside was hurt.
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.
The explosions killed 10 and wounded more than 30, said a Lebanese security official, speaking on condition of anonymity in line with regulations.
AP video showed firefighters extinguishing flames from vehicles, blood-spattered streets and bodies covered with sheets on the ground. A charred motorcycle stood outside the embassy gate.
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It was not immediately clear what had caused the blasts, and reports varied. Lebanon's state-run National News Agency said one of the blasts was carried out by a suicide attacker on foot while the second was a car bomb.
An armed guard of the Iranian 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 other explosion, which caused much more damage, was a car bomb, the guard said, speaking on condition of anonymity because he was not authorised to talk to media.
There have been several attacks the past few months, including with rockets and car bombs, against Shiite strongholds in Lebanon in what Syrian rebels say is retaliation for the militant Hezbollah group's open involvement on the side of President Bashar Assad's forces in the civil war in neighbouring Syria.