Little is known about the al-Ahvaziya group, but Islamic Revolutionary Guards Corps (IRGC) spokesperson Ramezan Sharif told ISNA it was "funded by Saudi Arabia".
Dozens of people, including women and children, were killed in an attack on a military parade in southwestern Iran on Saturday.
Gunmen disguised as soldiers in military uniforms, began shooting from behind the stands during the parade, according to the state-run news agency. The group provided no evidence for the claim, Reuters reported.
Iran is marking the anniversary of the beginning of the 1980-88 war with Iraq with several military parades across the nation.
Ahvaz is the capital of Iran's oil-rich Khuzestan province, which has seen attacks on oil pipelines by Arab separatists.
Iranian analysts said the attack may benefit the Guards.
The incident happened Saturday, in the country's southern region, bordering Iraq.
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"We stand with the Iranian people against the scourge of radical Islamic terrorism and express our sympathy to them at this bad time", Nauert said.
"The attack will strengthen the IRGC's position inside Iran and in the region", Tehran-based political analyst Hamid Farahvashian said.
But attacks on regime targets inside major cities are far rarer. It was the deadliest such attack in Iran in almost a decade.
The attack in the southwest city of Ahvaz saw four gunmen fire from a park near the parade into a crowd of marching Revolutionary Guardsmen, bystanders and government officials.
The British foreign ministry said its charge d'affaires had extended its condolences to Tehran and that Iranian officials were planning to lodge a formal complaint with Britain's media watchdog Ofcom.
Qassemi pointed to a bloody terrorist attack in the southwestern Iranian city of Ahvaz on Saturday, saying that the USA made baseless allegations against Iran while the Islamic Republic is one of the main victims of acts of terror by terrorist groups.
The National Resistance of al-Ahvaz, which describes itself as a general body comprising of a number of smaller groups, said it accepted responsibility for the attack, the BBC reported.
The Guard issued the statement a day after an attack claimed by Arab separatists in Ahvaz killed at least 29 people and wounded over 60. He told an Iranian opposition group that USA sanctions on Iran are leading to economic pain that could lead to a "successful revolution". To bolster his claim, he gave details about one of the attackers that the AP could not immediately verify. In June past year, suicide bombers attacked parliament and the mausoleum of the Islamic Republic's founder Ayatollah Khomeini, killing 18 people. Four attackers were involved and three of them have been killed by security forces who were on the scene and another was in custody in hospital after being wounded, Shekarchi said.
In a keynote speech, Rouhani vowed to boost Iran's ballistic missile capabilities despite Western concerns that were cited by his U.S. counterpart Donald Trump in May when he abandoned a landmark nuclear deal with Tehran.