It was reported to be the first time the Islamic State group has claimed an attack in Israel.
The jihadist group, in an online statement, said three IS fighters had targeted "a gathering of Jews" in an operation yesterday during which the three attackers were killed by Israeli police.
The group warned that "this attack will not be the last."
During the attack, the police officer was killed and others were wounded, according to IS.
It took place as Muslims marked the end of the third Friday of the fasting month of Ramadan, during which tens of thousands of Palestinians from east Jerusalem and the West Bank attended prayers at the nearby Al-Aqsa mosque compound, Islam's third-holiest site.
Israeli police said the three Palestinian assailants were killed.
According to police, two perpetrators opened fire at a group of police officers who returned fire, and a third stabbed the border policewoman a short distance away before being shot.
This was the first time that the Islamic State group has claimed responsibility for an attack in Israel, according to the SITE Intelligence Group, which monitors extremist organisations.
In its statement IS said the attack was "revenge for the religion of Allah and the sanctities of the violated Muslims."
Israel had eased restrictions on the entrance of Palestinians from the West Bank for Ramadan, including permitting daily family visits during Sundays through Thursdays.
Following the attack, Prime Minister Benjamin Netanyahu held a phone consultation with the chief of police and public security minister, and decided to cancel the family visits, a police statement read.
Netanyahu, however, did not revoke the permission given to Palestinian men aged over 40 from the West Bank to enter Jerusalem for Friday prayers, police said.
All three were from Deir Abu Mashal, a village near Ramallah, and had been arrested for or involved in "popular terror activity," a Shin Bet statement read.
A fourth Palestinian, a Hebron resident who had been identified by Palestinian security as a perpetrator, was in fact a passerby who was wounded by gunshots and taken to hospital, Israeli police said.
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