In a statement released hours after the attacks, the Revolutionary Guard said it held Saudi Arabia responsible for the Sunni jihadist group's actions and promised revenge.
In another statement, the news agency said the attacks on Khomeini's shrine were carried out by two suicide bombers.
Iran has been a key backer of Assad throughout Syria's six-year-long civil war.
The Sunni extremist group is at war with Iranian-backed forces in Syria and Iraq, and views Iran's Shiite majority as apostates deserving of death. The twin attacks wounded at least 42 people and killed 12.
Iran's Supreme Leader, Ayatollah Ali Khamenei, has repeatedly claimed the United States, Israel, and "the West" created the Islamic State to keep the Middle East destabilized.
Mobile phone lines were immediately disconnected, entrance and exit gates shut, and helicopters were seen circling the building moments after the initial attack.
State television described the shrine attackers as "terrorists" and said one had carried out a suicide bombing, without providing further details.
The source noted that the attackers opened fire at people who were visiting the sanctuary of the founding leader of the Iranian Islamic Revolution and injured several of them, while government officers arrested two of the attackers and the third one killed himself.
Efe quoted the Iranian Students News Agency (ISNA) as saying that one of the assailants was arrested while the other three remained holed up inside.
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At about 10:40 am Tehran time, gunmen and suicide bombers attacked at the mausoleum of Ayatollah Khomeini. The reports said Iran's security forces neutralized one terrorist, another one committed suicide. The Kremlin said the attack on two targets underlined the need for countries to pool their efforts to fight against terrorism, something it said meant working closely with Muslim nations.
Iranian TV later said the session of parliament had resumed, and broadcast footage of what it said was the opening session proceeding normally. Officials say the Islamic Revolutionary Guard Corps helped security forces control the situation.
An Associated Press reporter saw several police snipers on the rooftops of buildings around parliament.
Two teams of attackers used gunfire and explosives to strike Iran's parliament and the mausoleum of Ayatollah Khomeini in Tehran Wednesday, according to state media.
Islamic State has lambasted "heretic" Shi'ite Iran for helping the Syrian and Iraqi governments battle Islamic State, which considers Shi'ites to be infidels.
It was unclear how the gunmen who attacked the parliament building got past security checkpoints.
Also, the attack comes against a backdrop of regional tension, with Arab Gulf countries pushing for a more forceful isolation of Iran.
"Iranian officials will be called upon to step up intervention in Iraq (and) Syria big time", he said, adding that Wednesday's attacks will significantly boost IS morale amid battlefield defeats.
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