Five Americans who had been imprisoned in Iran were allowed to leave the country on Monday, President Biden said, after two years of high-stakes negotiations in which the United States agreed to unfreeze $6 billion in Iranian oil revenue and dismiss federal charges against five Iranians accused of violating U.S. sanctions.

The announcement that the Americans took off in a plane from Tehran just before 9 a.m. Eastern came as Mr. Biden and Ebrahim Raisi, Iran’s president, were to attend the annual United Nations General Assembly meeting of world leaders in New York on Tuesday.

The five Americans — some of whom had been held for years in Evin Prison, one of the most notorious detention centers in Iran — flew to Doha, the capital of Qatar, for a Cold War-style exchange with two of the five Iranians. Three others declined to return to Iran, according to U.S. officials.

In a statement, Mr. Biden said that “five innocent Americans who were imprisoned in Iran are finally coming home.” He added that they “will soon be reunited with their loved ones — after enduring years of agony, uncertainty and suffering.”

White House officials said the president held an “emotional call” with the families of the Americans.

The prisoner release was a critical breakthrough in a yearslong standoff over Americans who were imprisoned in Iran. But the terms of the deal have generated intense criticism from Republicans who say that releasing billions in oil revenue amounts to paying a ransom and would lead to the taking of more hostages.

The arrangement also comes as part of a larger effort by the Biden administration to de-escalate tensions with Iran, which soared in the years since President Donald J. Trump abandoned the 2015 nuclear deal that placed limits on Tehran’s nuclear program. But administration officials denied that the deal indicated a major shift in the long-hostile relationship between the United States and Iran.

Top aides to Mr. Biden have said financial sanctions and strict monitoring will prevent Iran from spending the money on anything except food, medicine and other humanitarian goods. They acknowledge, however, that the deal might free up money that Iran is already spending on those items for other purposes.

“Joe Biden’s embarrassing appeasement not only makes Iran stronger, it makes America less safe,” Senator Tom Cotton, Republican of Arkansas, wrote on X, the social media platform formerly called Twitter.

Biden administration officials have said the agreement with Iran was the only way to win the release of the five Americans, who the United States said had been wrongfully detained by the Iranians in deplorable conditions.

Siamak Namazi, one of the freed Americans, said in a statement that he had been dreaming of freedom for almost eight years as he experienced “torment” for 2,898 days in prison.

“My heartfelt gratitude goes to President Biden and his administration, which had to make some incredibly difficult decisions to rescue us,” he said.

The Americans — Mr. Namazi, Emad Sharghi and Morad Tahbaz, as well as two others who have not been named at their families’ request — had been jailed on unsubstantiated charges of spying. They had spent the last several weeks in Iran in home detention after Tehran agreed to release them from prison while the $6 billion transfer, a complicated process, was completed.

American officials said that Mr. Namazi’s mother and Mr. Tahbaz’s wife were also on the plane out of Iran. Both women are Americans and had been prevented from leaving the country.

The Americans were given a brief medical checkup in Doha before they boarded a U.S. government plane headed for Washington, officials said.

At the same time as the prisoner exchange, the United States informed Iran that it had completed the transfer of about $6 billion in Iranian oil revenue from South Korea to a Qatari bank account.

“This action was taken strictly to address a humanitarian need,” Mr. Raisi told journalists in New York on Monday. He added, “These were funds that belonged to the people of Iran.”

Mick Mulroy, a senior Pentagon official in the Trump administration, said on Monday that the release of the funds would “likely give countries that incarcerate Americans as political hostages more reasons to do so.”

Secretary of State Antony J. Blinken warned Americans on Monday against traveling to Iran and other countries where the risk of being wrongly detained is high.

“While this group of U.S. citizens has been released, there is no way to guarantee a similar result for other Americans who decide to travel to Iran despite the U.S. government’s longstanding warning against doing so,” he said.

The release took place two days after the first anniversary of the uprising in Iran that erupted after the death of 22-year-old Mahsa Amini in the custody of the country’s morality police. Hundreds were killed in the ensuing government crackdown, including at least 44 minors, and around 20,000 Iranians were arrested, the United Nations calculated. In the past few weeks, the government has arrested dozens of people to prevent a fresh round of protests.

“The international attention is now diverted from the ongoing horrific human rights situation in the country,” said Hadi Ghaemi, the executive director of the Center for Human Rights in Iran, a New York-based advocacy group. “To coincide with the anniversary of the uprising in Iran is seen as a slap in the face of Iranian people inside the country and has angered many.”

But officials at Iran’s mission to the United Nations dismissed the criticism, saying that the timing of the American detainees’ release was conditional on the $6 billion arriving in the Doha bank account and that Iran did not control that process.

Only some of the Iranians involved in the deal were jailed in the United States, though all of them faced federal charges. Those charges will be dropped under the terms of the deal.

Several of them are permanent residents of the United States. American officials said that two of the jailed Iranians decided to return to Iran on Monday. Iran’s foreign ministry spokesman, Nasser Kanaani, said that two would remain in the United States and one would return to a third country where he has family.

The Iranians were identified as Kaveh Afrasiabi, 65, who was charged with being an unregistered lobbyist; Reza Sarhangpour Kafrani, 48, a dual Iranian Canadian citizen charged with exporting lab equipment for Iran’s nuclear program; Mehrdad Ansari, who is serving a five-year prison sentence for obtaining military gear; Kambiz Attar Kashani, 45, a dual Iranian American businessman who pleaded guilty to conspiring to illegally export technologies; and Amin Hasanzadeh, who was charged with stealing sensitive technical plans.

Mr. Hasanzadeh has said he will return to Iran.

Negotiations to release the Americans from Iran accelerated in the spring, when Brett H. McGurk, the coordinator for the Middle East and North Africa at the White House, met with officials in Oman in early May.

In August, after Iran released the Americans to house arrest, U.S. officials said they would not celebrate until the Americans were out of Iran and on friendly soil.

The White House on Monday also announced new sanctions on Iran’s Ministry of Intelligence and its former president, Mahmoud Ahmadinejad, for wrongly detaining Americans in violation of the Levinson Act, named after Bob Levinson, a former F.B.I. agent who was believed to be held by Iran for years before his death.

The Biden administration has made considerable efforts during the last three years to win the release of Americans held in other countries.

In March, the United States secured the release of Paul Rusesabagina, a human-rights activist detained in Rwanda. In December, Russia agreed to release Brittney Griner, an American basketball star, in exchange for Viktor Bout, a convicted Russian arms dealer known as the “Merchant of Death.”

But other Americans remain in detention. In March, Russia accused the Wall Street Journal reporter Evan Gershkovich of espionage and detained him. Mr. Biden has said that his administration is working to secure Mr. Gershkovich’s release.

In his statement on Monday, the president said too many people were being held unjustly in Russia, Venezuela, Syria and elsewhere around the world.

“We remain unflinching in our efforts to keep faith with them and their families,” he said, “and we will not stop working until we bring home every American held hostage or wrongfully detained.”

Julian Barnes contributed reporting from Washington.