The assault on the parliament building appears to be over, after hours of intermittent, audible gunfire there. A suicide bomber died at the mausoleum.
Iranian officials say they managed to foil a third attack.
The so-called Islamic State group (IS) has claimed it carried out the attacks, which would be a first in Iran.
IS later posted a video which showed what it claimed was footage from inside the parliament building.
Iranian media reported that four attackers inside the parliament building had been killed by security forces.
It is not clear whether the death toll of 12 includes the attackers, or whether the victims were killed at both incidents or solely at the parliament.
About 40 people were injured in the two attacks, according to emergency services chief Pir Hossein Kolivand.
Gunmen armed with Kalashnikovs entered the parliament on Wednesday morning. Images from the scene showed a major security operation as forces surrounded the building.
Heavy gunfire could be heard.
Iranian authorities denied that there had been a hostage situation inside the parliament building.
Speaker Ali Larijani downplayed the events, describing it as a "minor issue".
Reports said the gunmen had entered parliament via a public entrance, dressed as women.
At about 10:40 (06:10 GMT) attackers at the mausoleum in southern Tehran, which is dedicated to the Islamic Republic's founder Ayatollah Khomeini, opened fire.
The governor of Tehran said one attacker there had detonated a suicide vest and another had been killed by security forces, state broadcaster Irib reports.
Images from the scene showed grenades and magazines for automatic weapons, apparently recovered from the body of an attacker.
The suicide attacker was a woman, reports suggested.
Several members of the public, visiting the shrine, have been injured.
Iran on Thursday began restoring internet access in the capital Tehran and a number of provinces, local news agencies and residents said, after a days-long nationwide shutdown meant to help stifle unrest over fuel price hikes.
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