Three Iranian-Americans arrived in Germany after leaving Tehran on Sunday in a prisoner swap that followed the lifting of most international sanctions on Iran under a deal US President Barack Obama said cut off Tehran's path to a nuclear bomb.

In a sign of sustained readiness to track Iranian compliance with remaining United Nations curbs, the United States imposed fresh sanctions on 11 companies and individuals for supplying Iran's ballistic missile programme.

The Obama administration had delayed the step for more than two weeks during tense negotiations to free five American prisoners, according to people familiar with the matter. Iran conducted a precision-guided ballistic missile test last October, violating a UN ban.

Speaking after the released Americans had left Iran, Obama said Iran now would not "get its hands on a nuclear bomb" and the planet would be more secure.

"This is a good day because once again we are seeing what's possible through strong American diplomacy," Obama said at the White House. "These things are a reminder of what we can achieve when we lead with strength and with wisdom."

Iranian President Hassan Rouhani hailed the nuclear deal as a "golden page" in Iran's history and said the agreement could be used as a model to resolve other regional issues.

The lifting of sanctions and the prisoner deal considerably reduce the hostility between Tehran and Washington that has shaped the Middle East since Iran's Islamic Revolution in 1979.

A Swiss plane took Jason Rezaian, the Washington Post's Tehran bureau chief; Saeed Abedini, a pastor from Idaho; and Amir Hekmati, a former US Marine from Flint, Michigan, as well as some family members, from Tehran to Geneva, Switzerland.

Shortly afterward, the three left for a US military base in Germany, arriving there later on Sunday, a US State Department official said.

One more Iranian-American released under the same swap, Nosratollah Khosravi-Roodsari, was not aboard the aircraft. A fifth prisoner, American student Matthew Trevithick, was released separately on Saturday, a US official said.

Several Iranian-Americans held in US prisons after being charged or convicted for sanctions violations have also been released, their lawyers told Reuters on Sunday.

"Doing a lot better"

Rezaian told two Post senior editors in a phone call on Sunday night that he was doing "a hell of a lot better than I was 48 hours ago".

The newspaper, which released details of the conversation with Rezaian, said he "found escape in the fiction he was allowed to read, and today he was avidly reading whatever he wanted".

Rezaian, 39, was arrested in July 2014 and sentenced in November to a prison term. Iranian prosecutors had accused him of espionage, charges the Post had dismissed as "absurd".

Obama called family members of the released prisoners on Sunday, including Rezaian's brother Ali, and Naghmeh Abedini, the wife of the Idaho pastor.

"I am thankful for our president and all of the hard work by the White House and State Department in making this happen," said Abedini, who has appeared with US Republican presidential candidate Ted Cruz, a US senator and a harsh critic of the Iran nuclear deal.

The American Iranian Council, which promotes better relationships between the United States and Iran, said in a statement on Sunday: "The prisoner exchange, Iran's dutiful implementation of its nuclear obligations, and the lifting of nuclear-related sanctions all herald a new era of US-Iran relations."

But the US thaw with Iran is viewed with deep suspicion by US Republicans as well as allies of Washington in the Middle East, including Israel and Saudi Arabia.

Cruz and fellow Republican presidential candidate Senator Marco Rubio praised Iran's release of five detained Americans on Sunday, but said the deal the White House made to win their freedom would lead to more Americans being taken "hostage."

Months of talks

The prisoner deal was the culmination of months of contacts, secret talks and legal manoeuvring that came close to falling apart on at least one occasion.

Speaking to parliament on Sunday, Rouhani, a pragmatic cleric elected in 2013 on promises to end Iran's years of sanctions and isolation, said he looked forward to an economic future less dependent on oil exports.

The exports are nevertheless likely to jump now that the United States, European Union and United Nations have scrapped the sanctions in return for Tehran complying with the deal to curb its nuclear ambitions, which Tehran says were peaceful.

But Rouhani noted bitter opposition to the lifting of economic curbs from Israel, some members of the US Congress and what he called "warmongers" in the region – an apparent reference to some of Iran's Gulf Arab adversaries, not least Saudi Arabia.

Presenting the draft budget for the next Iranian fiscal year, which begins in March, Rouhani told parliament the deal was a "turning point" for the economy of Iran, a major oil producer virtually shut out of international markets for the past five years.

He said later he expected five per cent economic growth in the next Iranian fiscal year and assured foreign investors of political and economic stability.

"The nuclear negotiations which succeeded by the guidance of the Supreme Leader and support of our nation, were truly a golden page in Iran's history," he said.

Tens of billions of dollars' worth of Iranian assets will now be unfrozen and global companies that have been barred from doing business there will be able to exploit a market hungry for everything from automobiles to airplane parts.

After the prisoners were freed, it was announced that the United States and Iran settled a long-standing claim, releasing to Tehran $400 million in funds frozen since 1981 plus $1.3 billion in interest, the State Department said. The funds were part of a trust fund once used by Iran to purchase military equipment from the United States, which was tied up for decades in litigation at the Iran-U.S. Claims Tribunal in The Hague.

End of sanctions era

In Tehran, ordinary Iranians were cautious about what the future holds after the lifting of sanctions. Many have lived under sanctions or wartime austerity for so long that they have no clear expectations about what the future might hold.

Iran's Gulf Arab adversaries were silent on news of the nuclear deal's implementation, in what was perhaps a sign of unease at the rapprochement.

Israel's opposition was evident in a statement from the office of Prime Minister Benjamin Netanyahu on Saturday night, which said that even after signing the nuclear deal, Iran had not yet "abandoned its aspirations to acquire nuclear weapons".

The International Atomic Energy Agency (IAEA) ruled on Saturday that Iran had fulfilled last year's agreement with six world powers to curtail its nuclear programme, triggering the end of sanctions.

Minutes after the IAEA's ruling, the United States formally lifted banking, steel, shipping and other sanctions on Iran. The EU likewise ended all nuclear-related economic and financial sanctions against the country. Most UN sanctions also automatically ended.

The end of sanctions means more money and prestige for Shi'ite Muslim Iran as it becomes deeply embroiled in the sectarian conflicts of the Middle East, notably in the Syrian civil war where its allies are facing Sunni Muslim rebels.