Three assailants were also killed following two simultaneous gun and knife attacks near the walled old city. Hamas rejects Daesh claim of responsibility.
Israeli security forces shot dead three Palestinians who carried out shooting and stabbing attacks in which an Israeli border policewoman was killed in Jerusalem on Friday, police said.
Daesh claimed responsibility for the attacks, the group's Amaq news agency said. But Hamas, the Palestinian resistance group that runs the Gaza Strip, dismissed the claim, saying the attackers had come from among its own ranks and the leftist Popular Front for the Liberation of Palestine (PFLP).
The attacks occurred simultaneously in two areas near the Damascus gate of Jerusalem's walled old city.
At one scene, two Palestinians were shot dead after opening fire at and trying to stab a group of Israeli police officers, police said. At the other, a Palestinian fatally stabbed a border policewoman before being shot dead by police.
A second Israeli officer was also injured in the attacks.
Israel's Shin Bet internal security agency identified the three assailants as Braa Salah, born in 1998, Adel Ankush, born 1999, and Asama Atta, born 1998, all from the village of Deir Abu Mashal near the West Bank city of Ramallah.
The PFLP said Salah and Atta had recently been released from several months in Israeli prison.
The names differ from those reported by Daesh, who said the attack was carried out by brothers called Abu al-Bara'a al-Maqdisi, Abu Hassan al-Maqdisi, and Abu Rabah al-Maqdisi.
A wave of Palestinian street attacks began in October 2015 but has since slowed. Israel blames the violence on incitement by the Palestinian leadership.
The Palestinian Authority, which exercises limited self-rule in the West Bank, denies that and says assailants have acted out of frustration over Israeli occupation of land sought by Palestinians in peace talks that have been stalled since 2014.