The milestone was welcomed by the parties involved in the deal, with U.S. Secretary of State John Kerry saying the world was a safer place as a result and Iran's President Hassan Rouhani hailing it as a "historic day" for the Islamic republic.
"We've now closed off every single path Iran had to building a (nuclear) bomb," U.S. President Barack Obama said Sunday. "We'll know if Iran ever tries to break out."
He added: "We've achieved this historic progress through diplomacy, without resorting to another war in the Middle East."
Speaking at a press conference in Tehran on Sunday, Rouhani hailed the agreement as "a tool for the progress, growth and development of the country, as well as the security and stability of the region."
He said that the agreement proved that "constructive engagement" with the world was the right path.
"As of today, it has become clear that the win-win policy is the right one," he said. "We can interact with the world in favor of our nation, and of course not in a manner that disfavors others."
Obama said that engaging with Iran had created a unique opportunity, and that the international community had reached a "milestone" in ensuring the Islamic republic did not develop a nuclear weapon.
'A new phase' in relations
Yukiya Amano, director general of the International Atomic Energy Agency, said that relations between Iran and the world body now entered "a new phase."
"It is an important day for the international community. I congratulate all those who helped make it a reality," Amano said.
The milestone meant that Iran had joined the UK, United States, France, Germany, China, Russia and the European Union in "the field of peaceful users" of nuclear energy, said Federica Mogherini, chief of foreign affairs for the EU.
Obama signed an executive order lifting the U.S. economic sanctions on Iran, the White House announced Saturday.
Kerry issued a statement confirming the IAEA had verified that Iran "has fully implemented its required commitments."
"Iran has undertaken significant steps that many, and I do mean many, people doubted would ever come to pass. And that should be recognized, even though the full measure of this achievement can only be realized by assuring continued full compliance in the coming years," Kerry said.
World 'safer' due to deal
UK Foreign Secretary Philip Hammond echoed Kerry's comment, saying the deal "makes the Middle East and the wider world a safer place." He added that Iran's nuclear program has been substantially rolled back.
In an address to Iran's parliament Sunday morning, Rouhani said that all the parties involved were satisfied with the deal, "with the exception of Zionists and warmongers ... American hardliners and extremists."
Israel was not one of the nations applauding the deal.
"Today, a country that threatens the existence of Israel, denies the Holocaust, destabilizes the Middle East, subjugates its own people and supports terror across the globe is being strengthened by the international community," said Yair Lapid, a member of the Knesset.
"The lifting of sanctions strengthens Hezbollah, it strengthens (Syrian President Bashar al-) Assad, it strengthens terrorists across the region who benefit from Iranian support."
Prime Minister Benjamin Netanyahu said Israel is monitoring Iran and will report any violation of the nuclear agreement. Iran has not given up on its ambition to acquire a nuclear weapon, he said.
The IAEA -- the U.N. nuclear watchdog organization -- released its report Saturday
assessing Iran's compliance with an agreement with foreign powers, including the United States and the European Union.
The release heralds "Implementation Day," the formal name for the start of the next phase in the agreement called the Joint Comprehensive Plan of Action, which was hammered out with Iran in July
. The new "Day" will mean the first wave of economic relief for Iran.
Under the agreement, Iran is obliged to take steps to put it further away from developing a nuclear weapon while keeping a peaceful nuclear energy program.
It must reduce its level of uranium enrichment, drastically reduce the size of its stockpile of enriched uranium, reduce the number of centrifuges and agree to unfettered international inspections.
Iran will be able to sell its oil again on world markets and its banks will be able to connect to the global system. The country will also regain access to billions in assets frozen overseas.
"As of today, our businessmen and -women, our entrepreneurs and traders can resort to the legal norms of banking to expand trade and exports," Rouhani told reporters Sunday.
"Today we are in an environment in which we can engage with the world in the economic, political and legal spheres to advance our national interest.
In a possible sign of the thaw in relations, it was announced that Iran had freed four American prisoners,
including Washington Post journalist Jason Rezaian, as part of a prisoner swap deal. Another American, who was not a part of the exchange, was also released.
New sanctions over ballistic missiles
But not all nuclear-related sanctions will be rescinded immediately -- that process won't be completed for about 10 years, should the deal hold.
And in a sign that the United States would continue to take action against Iran's ballistic missile program, the U.S. Department of the Treasury's Office of Foreign Assets Control announced Sunday that it had designated 11 entities and individuals it said were working on behalf of the program.
The office said in a statement that it had sanctioned the United Arab Emirates-based Mabrooka Trading Co. and an associated network, based in China and the UAE, that had been involved in procuring goods for Iran's ballistic missile program.
The network had done this by using front companies in third countries to obtain sensitive goods used in the production of ballistic missiles, disguising the end user of the goods, the statement said.
The office also sanctioned five Iranian citizens who it said had worked to procure ballistic missile components for Iran.
"Iran's ballistic missile program poses a significant threat to regional and global security, and it will continue to be subject to international sanctions," said Adam J. Szubin, acting under secretary for terrorism and financial intelligence.
"We have consistently made clear that the United States will vigorously press sanctions against Iranian activities outside of the Joint Comprehensive Plan of Action -- including those related to Iran's support for terrorism, regional destabilization, human rights abuses, and ballistic missile program," he said.