Such claims are often impossible to corroborate, and Islamic State has taken responsibility in the past for attacks in which it was not directly involved.
At least four other people were injured in the incidents, according to the emergency medical service.
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.
In its statement, IS said the attack was "revenge for the religion of Allah and the sanctities of the violated Muslims".
In a statement, police said that three Palestinians armed with knives and semi-automatic weapons were shot dead after carrying out two separate attacks near Jerusalem's Old City.
The attacks occurred simultaneously in two areas near the Damascus gate of Jerusalem's walled old city.
Three Palestinians carried out the attack.
Tens of thousands of Palestinians from the occupied West Bank were given permits by Israeli authorities to enter occupied East Jerusalem on Friday for Ramadan prayers.
Prime Minister Benjamin Netanyahu today accused the Palestinian president's party of praising Palestinians who killed an Israeli policewoman, calling for an end to payments to the families of attackers.
Israel had eased restrictions on the entrance of Palestinians from the West Bank for Ramadan, including permitting daily family visits during Sundays through Thursdays.
Israel, however, said it saw no evidence of such a decision, and the Palestinians wouldn't confirm. The measures were similar to those of previous years.