Three of the attackers were gunned down during clashes with the security forces and one other was arrested, news agencies reported.
NPR's Peter Kenyon reported that initially, attackers were described as "takfiri gunmen" by news agencies, a term sometimes used to describe Islamic State fighters.
Within an hour of this statement, Iran's Foreign Ministry summoned the United Arab Emirates charge d'affaires to rebuke him for comments made by an unnamed Emirati official.
Speaking on Sunday, Rouhani said: "America wants to cause chaos and unrest in our country so that it can return to this country, but these are unreal fantasies and they will never achieve their goals".
The Islamic State released a statement that claimed responsibility for Saturday's attack shortly after it occurred, though a separatist Arab militant group also claimed it was behind the attack.
The attack killed at least 25 people and wounded 60, according to the state-run IRNA news agency.
Rouhani did not identify those behind Saturday's attack, which was claimed by an Arab separatist group.
Foreign Minister Mohammad Javad Zarif said the attack had been carried out by "terrorists recruited, trained, armed & paid by a foreign regime".
The IRNA report said those wounded in the attack Saturday included a woman and a child. Iran's regional rivals, particularly Saudi Arabia and the UAE, have increasingly taken their decades-long behind-the-scenes pressure on the United States to bomb Iran into the open.
Tensions are running high after the attack with the U.S. and Gulf Arab states also being blamed by furious official in Tehran.
"The individuals who fired at the people and the armed forces during the parade are connected to the al-Ahvaziya group, which is fed by Saudi Arabia", Sharif said.
A number of Iranian officials, including Zarif and President Hassan Rouhani, are expected to travel to Trump's hometown of New York City next week for high-level meetings at the United Nations General Assembly.
Her words came just a day after the U.S. presidential lawyer, Rudy Giuliani, said that the Iranian government will be toppled.
"The United States condemns any terrorist attack anywhere, period".
Yacoub Hor al-Tostari, a spokesman for the Arab Struggle Movement for the Liberation of Ahvaz, told The Associated Press his group wants that area of southwest Iran to be its own nation.
Iran's Supreme Leader, Ayatollah Ali Khamenei, also called the attack a "continuation of the plots of the regional states that are puppets of the United States".
There was no immediate comment from Saudi Arabia on Rouhani's allegations.
The regional superpowers support opposing sides in the civil wars in Yemen and Syria as well as rival political groups in Iraq and Lebanon.
President Donald Trump's decision to withdraw the United States from an global nuclear deal from Iran will also give hardliners more political leverage because they argued that the United States should not be trusted.
IS claimed its first attack in Iran on June 7, 2017, when gunmen and suicide bombers hit the parliament in Tehran and the shrine of revolutionary leader Ayatollah Ruhollah Khomeini, killing 17 people and injuring dozens.
No group has yet claimed responsibility for the attack, but a spokesman for the Revolutionary Guards pointed the finger at ethnic Arab separatists.
Within hours of Saturday's attack, Iran summoned diplomats from Denmark, the Netherlands, and the United Kingdom, with IRNA reporting they were "informed of Iran's strong protests over their respective countries' hosting of some members of the terrorist group".