"Congratulations on this dignity and victory" as "we have opened a new chapter in our relations with the world", Rouhani said in a statement carried by the official IRNA news agency.
"Today is the era of a transition from sanctions to development," Rouhani was quoted as saying, adding that it requires work, innovation, investment, and grasping of the opportunities by all Iranians.
On Saturday, International Atomic Energy Agency (IAEA) Director-General Yukiya Amano said in a statement that: "Today, I released a report confirming that Iran has completed the necessary preparatory steps to start the implementation of the Joint Comprehensive Plan of Action (JCPOA)."
Iran also promises more transparency of its nuclear programme thus allowing the UN nuclear inspectors to monitor Tehran's atomic plan.
Iranian Foreign Minister Mohammad Javad Zarif and European Union foreign policy chief Federica Mogherini on Saturday jointly announced the implementation day of the Joint Comprehensive Plan of Action, the historic nuclear deal reached last July.
After the UN nuclear watchdog's announcement, the EU lifted its economic sanctions against Iran. Also, US President Barack Obama signed an executive order to lift sanctions on Iran related to its nuclear programme, said the White House.
"This is a historical moment... We have extended our friendship hands to the world and have opened new chapter in our relations with them," Rouhani said in the statement.
By the lifting of "oppressive sanctions ... we have crossed a dangerous turn in economy ... and our economy has been linked to that of the world."
The US and EU sanctions on Tehran's disputed nuclear programme in the past years had snapped the country's financial and economic ties with the international financial system and economic network.
"JCPOA is not a threat to any country. The friends of Iran are happy and the rivals should not be concerned. We are not a threat to any country. We give this message of peace, stability and security to the region and to the world as we protect Iran's dignity," the Iranian president said.
Earlier on Saturday, Iran and the US also swapped prisoners. Iran released four detained dual-nationals, including Iranian-American Washington Post reporter Jason Rezaian.
Also, US media reported Saturday that Washington would either pardon or drop charges against seven Iranians as part of the prisoner exchange.
The seven Iranians -- six of whom are dual citizens -- were either accused or convicted of violating US sanctions.
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