The attacks were the first claimed by ISIS inside tightly controlled Shi'ite Iran, one of the powers leading the fight against Sunni militants in the region.
Gorjizadeh said the terrorist incidents in Tehran showed the ignominy of the terrorists' mission, on the one hand, and the greatness of the Islamic establishment and the Iranian security forces, on the other.
Islamic State claimed responsibility for the suicide bombings and gun attacks on parliament and the mausoleum of the Islamic Republic's founder, Ayatollah Ruhollah Khomeini, on Wednesday.
The intelligence ministry on June 8 revealed the identities of the five attackers and said they had fought for the IS in Iraq and Syria.
After offering condolences to the families of the attacks victims, Alavi stressed that over the past few months the country's intelligence forces had neutralized over 25 terrorist cells.
The assailants attacked the parliament building in the morning.
Iranian lawmakers said the intelligence minister had assured them that "all Islamic state elements have been eliminated" in the country.
It added that the terrorists were detained in the Iranian provinces of Kermanshah, Kordestan, West Azarbaijan and Tehran.
IRNA news agency quoted a member of Majlis Presiding Board Akbar Ranjbarzadeh, as saying that Intelligence Minister Mahmoud Alavi and Interior Minister Abdolreza Rahmani Fazli as well as the IRGC Deputy Commander presented more details about the recent twin simultaneous attacks hitting the Iranian parliament and Mausoleum of Imam Khomeini in the south of Tehran, killing 17 and wounding 52.
On Saturday, the authorities also announced the arrests of eight people in Alborz province, north of Tehran, who allegedly provided logistical support to the attackers.
On Friday, Iran's supreme leader, Ayatollah Ali Khamenei, said the attacks would only increase Tehran's hatred against the U.S. and its Saudi Arabian "stooges".
Tehran police said the vehicle the attackers used on Wednesday was discovered on Saturday in the city centre.