Two of those injured were badly hurt, police said.
The attack occurred near police headquarters, close to the line dividing mainly Palestinian east Jerusalem from the mostly Jewish western sector of the city.
The area is also located near a light rail station. The assailant fled, but was pursued by police on motorcycles and was shot dead, police said.
The attacker had also opened fire on police, they said.
Details were still emerging from incident, which comes at a time of increased Jewish visitors to the flashpoint Al-Aqsa mosque compound in Jerusalem for the holidays of Rosh Hashanah, or Jewish new year, and Yom Kippur.
Last year's Jewish high holidays led to clashes and marked the start of an upsurge in Palestinian gun, knife and car-ramming attacks.
Violence since October 2015 has killed at least 231 Palestinians, 34 Israelis, two Americans, one Jordanian, an Eritrean and a Sudanese national, according to an AFP count.