The attack began when a gunman opened fire from his vehicle on civilians at a light-rail stop opposite the main police headquarters in Ammunition Hill, Israel Police foreign press spokesman Superintendent Micky Rosenfeld said.
Two of the victims, both women, were critically injured. One of them, 60-year-old Levana Melihi of Jerusalem, later died at Mount Scopus Hospital.
The attacker fled in the direction of the eastern Jerusalem neighborhood of Sheikh Jarrah, with police giving chase on motorcycles, police spokeswoman Luba Samri said.
When the attacker saw the officers in pursuit, he opened fire in their direction and police shot back, she said.
One officer, 29-year-old Yosef Kirme, was killed, and another wounded in the shootout.
Police described the shooting as a terrorist attack. The assailant, a 39-year-old Palestinian man from the east Jerusalem neighborhood of Silwan, was also killed, Rosenfeld said.
Authorities have not released the assailant's name due to an Israeli-issued gag order.
Hamas praised the attack and said the killer was one of its members. But it did not claim responsibility, suggesting the assailant acted on his own.
The perpetrator's family disputed Hamas' claim, saying he had no links to the Palestinian Islamist group that runs Gaza.
Attacker was due to be jailed
According to the assailant's family, he had been due to turn himself in to authorities Sunday to serve a four-month prison sentence stemming from a 2013 assault conviction.
The gunman was convicted of assaulting a police officer in the Old City of Jerusalem in 2013 and for spreading incitement on Facebook, according to a source familiar with the court rulings.
He was sentenced to 12 months in prison for incitement, and has been arrested and released two times in the last week, his family said.
The attacker was a member of Mourabitoun, an outlawed group of Muslim civilian guards that regularly confronts police and religious Jews on the al-Aqsa mosque compound, known to Jews as the Temple Mount. The group describes itself as the protector of al-Aqsa from Israeli settlers.
Following the attack, the perpetrator's brother was arrested, according to the family.
'Deplorable and unacceptable'
Rosenfeld said on Twitter that security had been heightened in Jerusalem to prevent further potential attacks.
"I condemn this morning's terror attack by a Palestinian perpetrator in occupied East Jerusalem," he said.
"Nothing can justify such attacks. My thoughts are with the families and friends of all victims and I hope for a full and speedy recovery of the wounded. It is deplorable and unacceptable that Hamas and others choose to glorify such acts which undermine the possibility of a peaceful future for both Palestinians and Israelis."
US State Department deputy spokesman Mark Toner echoed that sentiment.
"The United States condemns in the strongest possible terms the terrorist attack," he said in a statement.
"We extend our deepest condolences to the families of those killed and our hopes for a quick and full recovery for those wounded. There is absolutely no justification for the taking of innocent lives."