Soon after the assault on parliament, another bomber detonated a suicide vest near the shrine of the Islamic Republic's revered founder, Ayatollah Khomeini, a few kilometres south of the city, Zolfaghari said.
The Islamic State group has claimed responsibility for the attack.
Iranian commanders pointed the recent summit between Donald Trump and Gulf Arab leaders, where Saudi Arabia called for a harder line against Iran.
Putin said the attacks underscore the need for deeper worldwide cooperation to fight terrorism and confirmed Moscow's willingness to aid Iran.
In a statement after the attacks, the Guards said they "will never allow the blood of innocents to be spilt without revenge".
The IRGC has squarely blamed Saudi Arabia. "We offer our deep condolences to the people of Iran, government, and the bereaved families and pray for the speedy recovery of those injured". Before the security forces entered the building, ISIS' Amaq News Agency released footage of the attack.
State TV said one of the attackers was killed by security guards and that a woman was arrested following Wednesday's attack.
Lawmakers Elias Hazrati and Mehdi Kiaee confirmed an official report stating that the authorities resumed control of the hemicycle, surrounded by security forces.
Two gun shootings took place in Iran's parliament and Khomeini's Shrine on Wednesday, with some casualties reported.
"Some coward terrorists infiltrated one of the buildings of parliament".
Iran attacks leave 12 dead at parliament and Khomeini mausoleum
The IS group's Aamaq news agency released a 24-second video purportedly shot inside the parliament building during the siege. One of the assailants later blew himself up inside, according to a statement issued by Iran's state news agency, IRNA.
The group's militants are directly at war with Iranian-backed forces in Syria and Iraq, and view Shiites as apostates.
"The first terrorist team, comprising of two persons, entered the premises of the Imam Khomeini shrine at 10:30 (local time)".
Attacks of this nature are extremely rare in Iran, particularly in the highly-controlled capital where tourist and government sites are tightly policed.
The bill would impose mandatory sanctions on people involved in Iran's ballistic missile program and anyone who does business with them.
Saudi Foreign Minister Adel Al-Jubeir, speaking in Berlin, said he did not know who was responsible and there was no evidence Saudi extremists were involved. Four of them stormed the parliament in central Tehran, the capital, where heavy gunfire could be heard for hours. Eight people have been injured and one killed, according to Iranian state agency. "The guards became suspicious of their bags and when they wanted to search them, shooting began and they killed the security guard", he said.
Iran's counter-terrorism unit also said that one of the attackers at the shrine had been neutralized, in addition to the bomber.
Equipped with AK-47 assault rifles, handguns and suicide vests, the gunmen killed security guards and ordinary people before holding people hostage in the upper floors of the building.
"They (hardliners) are very angry and will use every opportunity to grow in strength to isolate Rouhani", said one of the officials.
Antonio Tajani, president of the European Parliament, said he "stands in solidarity with the President of Iranian Parliament and the Iranian people".
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