Some 13 people were hurt in the attack on Wednesday. The driver was shot dead.
Hamas militants said they carried out the attack. Israel's prime minister said it was a result of "incitement" by the Palestinian Authority and Hamas.
A similar car attack by a Palestinian took place in the same area two weeks ago which left a woman and a baby dead.
In a separate development on Wednesday, a Palestinian motorist drove into a group of soldiers in the southern West Bank, injuring three of them before fleeing the scene.
Israeli soldiers later found the car abandoned, according to witnesses at a nearby Palestinian refugee camp.
The driver turned himself in to security forces on Thursday morning, the Israeli army reported.
Meanwhile Jordan has recalled its ambassador to Israel over what it called the "unprecedented Israeli escalation" at holy and sensitive sites in Jerusalem.
As tensions have risen in the city, access to the Temple Mount/al-Haram al-Sharif compound has been intermittently restricted, with police at times barring male Muslim worshippers under the age of 50 from entering the site as a security measure.
The compound is known to Jews as the Temple Mount, the holiest site in Judaism, and to Muslims as al-Haram al-Sharif. It contains the al-Aqsa Mosque, the third holiest site in Islam.
Jordan also cited continued building of settlements by Israel in the West Bank and East Jerusalem, occupied by Israel in the 1967 Middle East war, and is to lodge a formal complaint to the UN Security Council in protest, reports say.
The settlements are considered illegal under international law, though Israel disputes this.
The man killed in Wednesday's van attack was police Chief Inspector Jidaan Asad, 38, police spokesman Micky Rosenfeld said.
The driver of the van - named as Ibrahim al-Akari - was from Shuafat refugee camp in the east of the city, police said.
His Facebook page states that he is a member of Hamas, and the Twitter account for the group's armed wing, the al-Qassam Brigades, described him as a member and a martyr.
Hamas, the militant Islamist group dominant in Gaza, and Palestinian President Mahmoud Abbas's Fatah party, agreed to form a unity government earlier this year - a move denounced by Israeli Prime Minister Benjamin Netanyahu.
Hamas rejects Israel's right to exist and advocates what it calls "armed resistance" against the Jewish state.
Mr Netanyahu said Wednesday's attack in Jerusalem was the result of incitement emanating from Mr Abbas and "his Hamas partners".
"We are in a prolonged battle in Jerusalem. I have no doubt we will win," Mr Netanyahu said.
"We are deploying all the necessary forces to restore calm and security to all parts of the city."
Hamas praised the attack as a "glorious operation", describing Mr Akari as a "hero" whose actions were a "natural response" to Israel's actions in east Jerusalem.
Two weeks ago a Palestinian from the Abu Tor area drove his car into a tram station, killing a baby and an Ecuadorean woman.
Holy site clashes
Earlier on Wednesday, dozens of masked protesters threw stones and launched fireworks at police near the non-Muslim visitors' entrance to the site, Israeli police said.
Police said they had managed to disperse the crowd.
The fate of Jerusalem is one of the most contentious issues between Israel and the Palestinians.
The Palestinians want East Jerusalem as the capital of a future state, while Israel regards the whole of Jerusalem as its "eternal and indivisible capital".
Israel occupied East Jerusalem in the 1967 Middle East war and annexed it in 1980 in a move not recognised internationally.