At around the same time in Sultan Suliman Street, a few hundred meters away, two Palestinians shot a Border Police force using a homemade Carl Gustav rifle.
Two of the Palestinians attacked police officers with guns and knives while the third stabbed the policewoman, police say.
The attack 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.
During the attack, the police officer was killed and others were wounded, according to IS.
Police alleged that the Palestinians shot dead were the assailants and from the West Bank area and. One of the attackers was reportedly holding a gun, but it jammed. The third was said to be a 30-year-old resident of Hebron.
Two assailants attacked a group of officers with knives on Friday at the Damascus Gate, The Times of Israelreported.
Medics said four people besides the officer were wounded in the incident, two of them moderately and two lightly.
The Islamic State has launched its first attack in Jerusalem, with the claim of responsibility nearly immediately following an attack that left one Israeli police officer dead.
The Islamist movement Hamas that rules the Gaza Strip condemned the shooting of the Palestinians as a "crime by the occupation".
The jihadist group in an online statement said three IS fighters had targeted "a gathering of Jews" in an operation Friday, and warned that "this attack will not be the last".
More than 272 Palestinians, 42 Israelis, two Americans, two Jordanians, an Eritrean, a Sudanese and a Briton, have been killed in the occupied West Bank since October 2015, according to an AFP tally.
Israel blames the violence on incitement by Palestinian political and religious leaders.
Israeli authorities, meanwhile, say nearly 50 Israelis have been killed in attacks carried out by Palestinians over the same period.