Iran's ambassador to Lebanon, Ghazanfar Rokn-Abadi, was not hurt in the blast, but he said Iran's cultural attache, Sheik Ibrahim Ansari, was killed.
The embassy building and many vehicles and apartments along the street were also damaged.
A Beirut correspondent for RFE/RL's Radio Farda at the scene of the blasts, Ilia Jazayeri, described massive damage to the surrounding area and nearby buildings.
He also saw many injured people and relatives looking for family members at the scene, with dead bodies still lying on the street.
The embassy is located in a part of the Lebanese capital that is a stronghold of the Shi'ite movement Hizballah, which, along with Iran, supports Syrian President Bashar al-Assad in the country's civil war against rebels.
Initial reports say two suicide bombers set off the explosions.
A guard at the Iranian Embassy said a man drove up on a motorbike and then detonated his explosives, which blew down the embassy's main entrance gate.
Shortly afterward, a second explosion -- reportedly from a vehicle laden with explosives -- went off.
A Lebanese-based Al-Qaeda-linked group known as the Abdullah Azzam Brigades has claimed responsibility for the attack.
Sirajeddin Zreikat, a member of the group, wrote on his Twitter account that "this is a double martyrdom operation carried out by two heroes from the heroic Sunnis of Lebanon."
But Iranian Foreign Ministry spokeswoman Marzieh Afkham said the attack is "an inhumane crime and spiteful act done by Zionists and their mercenaries."
U.S. Secretary of State John Kerry issued a statement strongly condemning what it called "senseless and despicable terrorist bombings."
Russia's Foreign Ministry said in a statement that it deplored the "bloody attack on a diplomatic mission."
The November 19 blasts follow two other bombings this year in Beirut's southern suburbs, which are dominated by Hizballah. Many suspect Syrian rebel groups are behind the attacks as retaliation for Hizballah's support for Assad.