Daoud and rights groups said she had submitted an application several months ago to reenter Israel to retrieve baby Maryam from a hospital in Israeli-annexed east Jerusalem, but received no reply.
The Israeli defence ministry body responsible denied that any application had been received.
Palestinians in the Gaza Strip need to apply for individual permits to enter or transit Israel, but do so through the Palestinian Authority.
The PA circulated a document allegedly showing that the request had been submitted.
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Tiny Maryam smiled as she was picked up by her mother for the first time since her premature birth on August 1.
"It's a very beautiful feeling. Finally I can take her in my arms. Now I just hope that she'll be happy for ever," her grinning mother said.
Daoud said the Israeli authorities called late Sunday to say she could cross the Gaza-Israel border, shortly after a television feature on the case.
"She had no maternal affection, nobody held her," said a tearful Umm Abdul Rahman, a woman from east Jerusalem who said she came in weekly to check on Maryam, sending photos to the family via WhatsApp.
Daoud was previously granted permission to travel to Jerusalem in July 2016 for medical treatment as there were complications with her pregnancy.
Hospitals in Gaza do not have the facilities to care for premature babies, said paediatrician Mohammed al-Najjar at Makassed Hospital in east Jerusalem where Maryam was born more than two months early along with a sibling who died.
She needed to be kept in an incubator for several weeks in order to survive, doctors said.
Daoud said she now had to return to Gaza to care for her other two young children, assuming she could do so quickly once Maryam's health improved.
Her story was covered by Israel's Channel 10 on Sunday night, and the permit was granted shortly afterwards.
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