An armed civilian killed the driver, 39-year-old Hatem Najm, police confirmed after the incident.
All those injured during the incident were taken to the same hospital.
The condition of the 70-year-old man was serious at the time. The 30-year-old woman was in moderate condition, while the three men, one in their 50s and two in their 25s, suffered only minor injuries.
Najma, a married father of five, was not known to Israeli security forces but had a history of mental illness.
The attack took place in central Jerusalem, a block away from the busy Mahane Yehuda outdoor market.
Police say Najma was speeding at the intersection and intentionally ran into a group of pedestrians crossing the street at the intersection. He was then shot by a gunman while still behind the wheel.
Israel’s Day of Remembrance, dedicated to commemorating the lives of fallen soldiers, begins Monday night.
Israeli President Benjamin Netanyahu called the incident a terrorist attack.
“A few minutes ago there was another attempt to kill Israeli citizens not far from here. This terrorist attack reminds us that the Land of Israel and the State of Israel were won through many hardships,” Netanyahu said earlier in a Memorial Day ceremony.