The UN's atomic watchdog late on Saturday confirmed that Iran had complied with its obligations under last summer's accord and the United States and European Union announced they were lifting the sanctions that have for years crippled the country's economy.
Western governments hailed the announcement as a milestone though some critics, including Israel, alleged that Tehran was still seeking to develop a nuclear weapon.
The announcement followed news of a prisoner swap between Iran and the United States that will include the release of Washington Post correspondent Jason Rezaian, in another sign of thawing relations between the longtime foes.
Rouhani, a moderate whose 2013 election victory helped launch a huge diplomatic effort toward the deal struck on July 14 in Vienna, said implementation of the agreement did not harm any country.
"We Iranians have reached out to the world... have opened a new chapter in the relations of Iran with the world," the official IRNA news agency quoted Rouhani as saying.
Iran is "not a threat to any government or nation". The agreement, he said, "is not a loss for any country".
Rouhani, who has promised that 2016 will be a "year of prosperity" for Iranians, was to give a press conference later Sunday.
Addressing lawmakers in parliament he said Iran, no longer burdened by sanctions, would seek foreign investment of $30-50 billion annually, to dramatically spur growth to eight percent.
EU foreign policy chief Federica Mogherini, representing the six world powers, announced the lifting of "multilateral and national" sanctions in Vienna, saying the deal showed that intense diplomacy could resolve even "the most difficult issues".
KERRY SAYS WORLD IS SAFER
The United States lifted a raft of sanctions, with Secretary of State John Kerry saying in Vienna: "The United States, our friends and allies in the Middle East, and the entire world are safer because the threat of the nuclear weapon has been reduced."
Iran can now resume widespread oil exports, long the lifeblood of its economy though Rouhani has moved away from relying on crude. Less than 25 percent of revenues in this year's budget will come from oil.
The nuclear deal will also open up business in the 79-million-strong country.
The Vienna agreement was nailed down after two years of rollercoaster negotiations following Rouhani's election.
The highly complex deal drew a line under a standoff dating back to 2002 marked by failed diplomatic initiatives, ever-tighter sanctions, defiant nuclear expansion by Iran and threats of military action.
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In addition the nuclear talks put Iran and the United States on the road to better relations, more than three decades after the Islamic revolution that toppled the US-backed shah.
Four of the five detainees to be freed by Tehran under the prisoner swap, including Rezaian, were still in the country as of early Sunday, with a US official saying "logistical steps" were ongoing.
Under the exchange, Washington said it had granted clemency to seven Iranians, six of whom were dual US-Iranian citizens, and dropped charges against 14 more.
International Atomic Energy Agency chief Yukiya Amano was also due in Tehran on Sunday for talks on the UN watchdog's enhanced inspections to ensure Iran's continued compliance with the deal.
The steps taken so far by Tehran extend to at least a year -- from a few months previously -- how long Iran would need to make one nuclear bomb's worth of fissile material.
They include slashing by two-thirds its uranium centrifuges, reducing its stockpile of uranium -- enough before the deal for several bombs -- and removing the core of the Arak reactor which could have given Iran weapons-grade plutonium.
Iran has always denied wanting nuclear weapons, saying its activities are exclusively for peaceful purposes including power generation and medical research.
ISRAEL: TEHRAN STILL WANTS BOMB
Critics, including US President Barack Obama's Republican opponents, have poured scorn on the deal, saying it fails to do enough to ensure Iran will never acquire the bomb.
Israel, widely assumed to be the Middle East's only nuclear-armed state and Iran's arch-foe, has repeatedly slammed the agreement.
Israeli Prime Minister Benjamin Netanyahu said Saturday Iran "has not relinquished its ambition to obtain nuclear weapons."
And on Sunday he told his cabinet: "Israel's policy has been and will remain exactly what has been followed: to not allow Iran to acquire nuclear weapons."
Sunni Saudi Arabia, Iran's main regional rival, is also alarmed at the prospect of warmer US-Iran ties and of predominantly Shiite Iran, increasing its influence.
Tensions between Saudi Arabia and Iran, already fighting a proxy war in Yemen and backing opposing sides in the Syrian conflict, have reached new heights in the past two weeks.
Saudi Arabia and a number of its Sunni Arab allies cut diplomatic ties with Tehran after protesters angry at Riyadh's execution of a prominent Shiite cleric on January 2 sacked its embassy in Tehran.
Iran's imminent return to the oil market has also contributed to the sharp slide in the price of crude to 12-year lows of under $30 per barrel this week, putting public finances in Gulf nations under strain.
All seven Gulf bourses tumbled on a wave of panicked sell-offs on their first trading day of the week on Sunday.
The Saudi Tadawul All-Shares Index, the largest Arab market, fell 7.2 percent to below 5,500-points during trading. The level was last seen in early 2011.