The ministry issued a statement on its website with bloody pictures of the men's bodies.
The statement only released the first names of the terrorists and said that their last names were being withheld due to "certain social and security considerations".
The statement described the five men as "long affiliated with the Wahhabi", an ultraconservative form of Sunni Islam practised in Saudi Arabia.
"The five known terrorists. after joining the IS terrorist group, left the country and participated in crimes carried out by this terrorist group in Mosul and Raqqa", the ministry said.
"Last year, they returned to Iran.to carry out terrorist attacks in the holy cities of Iran", the ministry continued.
State TV said one of the attackers was killed by security guards and that a woman was arrested following the shrine attack.
Iranians, contacted by Reuters, said there was a heavy police presence, particularly in the capital's busy squares. "So people in Iran, they don't consider the U.S. as their enemy".
The death toll from Wednesday's attacks on the Iranian parliament building and the mausoleum of Imam Khomeini, claimed by Daesh (ISIL) terrorist group, grew to 17 on Thursday.
USA - and Iran-backed forces are locked in a race to take Islamic State strongholds in southeastern Syria and seize a stretch of land that will either cement Tehran's regional ambitions or stifle them.
NPR's Michele Kelemen notes that the State Department condemned the attack and wrote that "the depravity of terrorism has no place in a peaceful, civilized world".
Trump said the United States would "grieve and pray" for the victims, but added: "We underscore that states that sponsor terrorism risk falling victim to the evil they promote".
Woman charged with top-secret leak denied bond
Neither prosecutors nor Winner's parents have identified the government agency where she worked at the time of her arrest. FILE - In this June 6, 2013 file photo, the National Security Agency (NSA) campus in Fort Meade, Md.
Iran's foreign minister on Wednesday called the White House statement about the double attack on Tehran that killed 16 people "repugnant" after it linked the assault to what it called Iran's support for terror. "Iranian people reject such USA claims of friendship", he wrote on Twitter.
In a statement, the IRCG called it "meaningful" that the attacks followed President Donald Trump's visit to the Sunni-ruled kingdom, where he strongly asserted Washington's support for Riyadh.
Iran's elite Revolutionary Guards have pointed the finger at regional rival Saudi Arabia - a close USA ally - which Tehran accuses of funding extremism and groups including IS.
The "spilt blood of the innocent will not remain unavenged", the Revolutionary Guard added.
The House also adopted another unanimous resolution expressing deep concerns over the recent developments in relations involving brotherly Muslim states in the Gulf region including the Kingdom of Saudi Arabia and Qatar.
"If Iran had not confronted terrorists where the core of this sedition is, it would have faced more attacks in Iran", he said. His message is believed to be directed at Saudi Arabia.
The Islamic State group claimed the Tehran attacks; no link has emerged of Saudi Arabia's involvement. They never named the country directly, but the implication was clear.
Claiming the attack, IS posted a video which showed what it claimed was footage from inside the parliament building.
In the attack at the mausoleum, one suicide bomber blew himself up while the second one was killed in a gunfight, according to the semi-official Tasnim news agency.
"She helped others, sent them to their missions after the start of the attacks, a member of the Iranian committee of national security of the parliament told Mehr news agency". The revered shrine was not damaged.
The attack appears to be inflaming the enmity between the countries.
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