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.
"Female border policewoman injured critically in attack at Damascus gate," police spokesman Micky Rosenfeld wrote in a statement. "3 Arab terrorists shot by police units that responded at the scene."
Police said the three were killed.
The policewoman was taken to hospital in critical condition and later died of her wounds. She was identified as Hadas Malka, 23, a staff sergeant major.
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.
Jerusalem police chief Yoram Halevy identified the three attackers as Palestinians from the West Bank.
Ramadan activity
Israel had eased restrictions on the entrance of Palestinians from the West Bank for Ramadan.
"During Ramadan there are large numbers of (Palestinian) youths who enter without permits, they take advantage of Ramadan to be in Jerusalem," Halevy told media at the scene of the attack.
The area around Damascus Gate was sealed off in the hours after the attack, with a few youths throwing fireworks at security forces.
A road leading to Damascus Gate full of stalls opened especially for Ramadan was closed off by police, and a shopkeeper said this would normally be one of the busiest nights of the year.
Inside the Old City shops were open but the atmosphere was subdued and numbers of people much lower than normal.
Large numbers of heavily armed security forces were patrolling throughout the Old City, an AFP reporter said.
The Shin Bet internal security agency identified the three perpetrators as Braa Salah, born in 1998, Adel Ankush, born in 1999, and Asama Atta, born in 1998.
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 for care, Israeli police said.
Medics said four people besides the officer were wounded in the incident, two of them moderately and two lightly. Two of those wounded were Palestinians from east Jerusalem.
The Islamist movement Hamas that rules the Gaza Strip condemned the shooting of the perpetrators as a "crime by the occupation."
A spokesman for the movement, Hazem Qassem, said the attack was "proof that the Palestinian people are leading a revolution against the enemy."
He also said the attack proved the Palestinian Authority's policy of security coordination with Israel was wrong, calling it "a national crime".
A wave of unrest that broke out in October 2015 has claimed the lives of 272 Palestinians, 42 Israelis, two Americans, two Jordanians, an Eritrean, a Sudanese and a Briton, according to an AFP tally.
Israeli authorities say most of the Palestinians killed were carrying out knife, gun or car-ramming attacks.
The Old City in annexed east Jerusalem is one of the focal points of the wave of violence.
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