"We grieve and pray for the innocent victims of the terrorist attacks in Iran, and for the Iranian people, who are going through such challenging times", Mr Trump's statement read.
"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, adding that in 2016 they returned "to carry out terrorist attacks in the holy cities of Iran".
IS has long vowed to attack Iran because the country had deployed military advisers and support to both Syria and Iraq in their fights against the extremist group.
Foreign Minister Javad Zarif also pointed a finger at Riyadh, though he did not call the country by name Wednesday night.
Head of Strategic Research Center Ali Akbar Velayati voiced his condolences over the martyrdom of a number of Iranian people in near-simultaneous attacks on Iran's Parliament and the mausoleum of Ayatollah Khomeini, saying despite the tragic event, the great Iranian nation is standing together against enemies' plots.
Mohammad Javad Zarif wrote on his Twitter account: "Repugnant White House statement".
On Thursday, Iran's Intelligence Minister Mahmoud Alavi said investigators were working to determine whether Saudi Arabia had a role Wednesday's attacks but said it was too soon to say if that was the case.
"Iranian people reject such US claims of friendship", Zarif tweeted.
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Iranian women light candles in Mother Square in Tehran in 2001 in memory of victims of the September 11 terrorist attacks.
Iran's powerful Revolutionary Guard indirectly blamed Saudi Arabia for the attacks. The attackers started shooting visitors of the shrine. Commuters in the Iranian capital noticed more police than usual on the streets as dawn broke.
Emirati Minister of State for Foreign Affairs Anwar Gargash told The Associated Press on Wednesday that "the Iranian government should not use the attack in a very polarized situation against Saudi Arabia or claim that Saudi Arabia is somehow linked to the attack, because it isn't".
Larijani also criticized a Wednesday decision by the U.S. Senate to move forward on a new set of sanctions against Iran, including its elite Revolutionary Guards, a move that came on the same day as the Tehran attacks.
The violence began in midmorning when assailants with Kalashnikov rifles and explosives stormed the parliament complex where a legislative session had been in progress. At least two of them blew themselves up with suicide vests, Iranian media reported. One showed a toddler being handed through a first-floor window to safety outside as an armed man looked on.
The speaker claims to represent the "first battalion" of IS formed inside of Iran. The revered shrine was not damaged.
Tehran's stock exchange fell almost 2 percent Thursday after the attacks.
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