Last Friday, Abigail Edan turned 4. The little girl spent her birthday in captivity, held hostage by Hamas.

On Oct. 7, she watched her mother, Smadar, get shot and killed by Hamas militants who broke into their home. Her father was shielding her but was gunned down moments later. She crawled out from under his body and sought refuge at a neighbor's house. But militants found her there and took her hostage along with the family of five living next door, her relatives said.

For 50 days, Abigail's family did not give up hope that the child - whom they have described as a precocious toddler who "ran" her household - would return home.

On Sunday, their hopes came true. Abigail, a dual citizen of Israel and the United States, was the first American hostage freed by Hamas as part of a deal struck with the Israeli military that the Biden administration helped broker amid the ongoing war. She was the youngest of the 10 Americans believed to be taken hostage after the Oct. 7 attack by Hamas on Israel.

"We hoped and prayed today would come," Abigail's cousin, Noa Naftali, and her great-aunt, Liz Hirsh Naftali, said in a statement Sunday after her release. "There are no words to express our relief and gratitude that Abigail is safe and coming home."

On the day of the attack, Abigail's family was in the home they shared on Kibbutz Kfar Azza, about two miles from the Gazan border. Her father, Roy Edan, a photographer for the Israeli news outlet Ynet, had been taking some of the earliest photographs of Hamas's attack on Israel, according to his employer.

After militants rushed their home, the two oldest children, ages 6 and 10, locked themselves inside a closet and hid for 14 hours. They survived and were reunited with family soon after the attack.

Abigail, however, was missing.

Members of the extended family initially thought Abigail had also been killed alongside her parents. But it later became clear that the little girl had been captured by Hamas.

Before Sunday's release, Liz Hirsh Naftali told The Washington Post that the thought that Abigail would not spend her birthday with her siblings was devastating.

The prospect of recovering their sister, Liz Hirsh Naftali told CBS News in an interview, was the elder children's "one hope."

"Yes, they witnessed it. They are 6 and 10. They understand," she said of Abigail's siblings. "The one piece that they can only hope that will come to give some sort of an ability for a closure and to move on is that Abigail comes back."

She described Abigail as an energetic, lively child who "was always in the front."

Now, Abigail will be reunited with her siblings and live with their aunt, uncle and grandparents in Israel, Noa Naftali told CBS News last week. They would be able to offer "the love and support that they need after losing their parents."

Soon after Abigail's release on Sunday, her grandfather Carmel Edan told reporters outside his home that he could not believe his eyes when he first saw photos of his granddaughter en route to Israel.

"It was just wow. I didn't believe it until I saw it," he told reporters outside his home, according to the Times of Israel.

On Sunday, when asked what is the first thing he will say to Abigail when he sees her, Carmel Edan said he will first kiss, hold and hug his granddaughter.

"Speak? No," he said. "There's nothing to say."

The children's grandfather said he has not yet said anything about her release to her siblings. The two, he said, are "in shock and with a psychologist." The wound of losing their parents will "never heal."

"Now I'm calm, but not completely calm, because there is happiness, but there is also the absence of [Roy and Smadar]," he said.

President Joe Biden first announced Abigail's release around noon Sunday. Speaking in Nantucket, Mass., the president said the girl "has been through terrible trauma."

"Today she's free," he said from the White Elephant Hotel. "Thank God she is home."

In their statement Sunday, the Naftalis thanked the White House team, as well as the Qatari government and "other informal actors who are involved in securing Abigail's release and reuniting other hostages with their loved ones."

"We can get all hostages back home. We have to keep pushing," the Naftalis added.

Abigail's grandfather, while speaking to reporters outside his home, also thanked Biden.

"We love him very much for all the help he is giving us, to Israel," he said. "Keep going. Keep going."

In his remarks Sunday, the president acknowledged that "innocent children in Gaza are suffering greatly as well." According to the Gaza Health Ministry, more than 13,300 people have been killed in Gaza and 35,180 have been wounded since the Israel-Gaza war began.

The president also said he would remain engaged in trying to secure the release of other Americans, and noted that he would continue working with the governments of Israel, Egypt and Qatar.

"The proof that this is working and worth pursuing further is in every smile and every grateful tear we see on the faces of those families who are finally getting back together again," Biden said. "The proof is little Abigail." Biden also spoke with Abigail's family in Israel and the United States after her release.

Officials believe that eight other U.S. citizens and one U.S. green card holder are still being held captive by Hamas. Two American women - a mother and daughter from Evanston, Ill. - were released Oct. 20 after 13 days in captivity.

Biden on Sunday spoke of the importance of the deal that his administration helped negotiate, adding that he hoped it could remain in place until all the hostages are released.

"I'm hopeful this is not the end, that it's going to continue. But we don't know," he said.

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Joanna Slater and Matt Viser contributed to this report.