The Americans imprisoned by Iran began their journey home Sunday, their friends and family awaiting emotional reunions, after delicate diplomatic negotiations that played out quietly in the shadows of international nuclear talks.

“This is a good day,” President Barack Obama said from the White House. “When Americans are freed and returned to their families, that’s something we can all celebrate.”

A charter plane left Tehran for Switzerland with the Americans — all four who had been detained, according to Iran state television, or only three, the U.S. said — as part of a prisoner swap.

U.S. officials said Washington Post reporter Jason Rezaian, former U.S. Marine Amir Hekmati, and pastor Saeed Abedini were on the flight, but not Nosratollah Khosravi-Roodsari. But Iran said he had taken off, too. The discrepancy could not immediately be reconciled.

“Those who wished to depart Iran have left,” according to a senior U.S. official, speaking on condition of anonymity because of the sensitivity of the matter and privacy concerns for the families.

The Post’s publisher, Frederick J. Ryan, Jr., said in a statement, “We are relieved that this 545-day nightmare for Jason and his family is finally over.

Hekmati’s family said “it is hard to put into words what our family feels right now.”

Iran celebrated the lifting of stiff economic penalties now that the Islamic Republic has met a critical benchmark as part of the agreement to pull back its nuclear program. Also, a dispute dating to before the 1979 Islamic Revolution of a military purchase by Iran was finally settled, sending $400 million to Tehran, plus $1.3 billion in interest.

But not all news was good for Tehran: The U.S. on Sunday imposed new sanctions against 11 individuals and entities involved in Iran’s ballistic missile program as a result of Tehran’s firing of a medium-range ballistic missile in October.

U.S. officials have said the Treasury planned to announce the penalties in late December, but held off after Iran’s foreign minister said they could have derailed the prisoner exchange that took place this weekend.

Secretary of State John Kerry said U.S. officials hashed out the prisoner exchange over 11 or 12 meetings with the Iranians. At times, the Americans thought a deal was set, only to get stuck on the details.

After almost constant conversation over the last few days, they finally did settle it.

Kerry said the nuclear agreement provided the key impetus.

Just before Iran’s foreign minister, Mohammad Javad Zarif, announced the July 14 accord with his European Union counterpart, Kerry again raised the issue of the detained Americans.

A photograph of him speaking with Zarif and Iranian President Hasan Rouhani’s brother, Hossein Fereydoun, captured the moment.

Things progressed significantly by a November meeting in Vienna on Syria’s civil war, when, for the first time, Iran was included in the discussions. Kerry and Zarif met on the sidelines of the talks to discuss the prisoners.

“We actually shook hands thinking we had an agreement,” Kerry said. “I thought it was done.”

But the deal bogged down in Tehran and never went through. “So we went back to work,” Kerry said.

He described the negotiations as difficult, especially as the Iranians made what he said were unacceptable demands. Kerry said the United States made it clear that it would not release a hardened criminal such as an accused murderer or someone with a narcotics record.

“For a long time, this didn’t move because of the people they were asking for,” Kerry recalled. “We said, ‘No, and no, and no.’”

“And believe me, it’s hard when somebody says to you, ‘Hey, you give us this guy, we let them all out.’ And you have to say no. And you know you’re keeping people in a not very nice place for the next whatever number of months,” he said.

“But there have to be an enforcement of our principles and our standards here. And in the end, we came out in the right place on that.”

More progress was made by Kerry’s meeting with Zarif on Dec. 18 in New York. By then, American and Iranian teams in Geneva were working hard on the details of the swap.

The U.S. was prepared to release individuals who violated nuclear-related sanctions on Iran, Kerry said. By Saturday night, those penalties were no longer in effect.

“In the end, the president made the call,” he said.

Talking to reporters in his plane after landing Sunday morning at Andrews Air Force Base, Kerry said he had hoped to meet the returning Americans in Switzerland.

But after the nuclear talks moved to Vienna and dragged on, he decided to send the lead negotiator on the matter, Brett McGurk, and State Department aide Patrick Kennedy, to await the Americans’ arrival.

One of the last hiccups that delayed the Americans’ departure was an Iranian military official’s misunderstanding about Rezaian’s wife and mother joining him on the flight. After Kerry spoke to Zarif, that problem was solved. Permission was granted.

But the various administrative holdups meant that the Swiss crew set to fly the plane ran into a mandatory crew rest. That set back takeoff several hours.

Kerry said the prisoner swap and almost simultaneous implementation of the nuclear deal raise the prospects of increased U.S.-Iranian cooperation on other matters.

Zarif, Kerry said, made it clear that if they got the two tasks done, “there are ways to try to translate this and hopefully be constructive in other things. He specifically said Syria and Yemen.”

“I put a big, ‘Who knows?’” on that, Kerry said, but expressed hope.

Kerry said he would remain at work on other Americans still being held in Iran.

The exchange did not cover Siamak Namazi, an Iranian-American businessman who advocated better ties between Iran and the U.S. and who reportedly was arrested in October, or former FBI agent Robert Levinson, who disappeared in Iran in 2007 on an unauthorized CIA mission.

Asked if Levinson was dead or alive, Kerry said, “We have no idea.”

The exchange eases a leading irritant as the two countries gingerly explore prospects for a smoother relationship after decades of hostility — even as they remain sharply at odds on other fronts.

A fifth American, student Matthew Trevithick was released independently of the larger swap, and already had headed home.

In turn, the U.S. was pardoning or dropping charges against seven Iranians — six of whom are dual U.S.-Iranian citizens — accused or convicted of violating U.S. sanctions.

Three were serving prison terms and received a commutation or pardon. Three others were awaiting trial; the last one made a plea agreement.

In addition, the U.S. was dropping drop Interpol “red notices” — essentially arrest warrants — on 14 Iranian fugitives it has sought, officials said.