Israeli police said it was a terrorist attack and that the driver was shot and killed.
The Magen David Adom ambulance service said four 20-year-olds - three women and a man - had been pronounced dead.
"It is a terrorist attack, a ramming attack", a police spokeswoman said on Israel radio.
At least 15 other people were wounded when a truck veered off the road and rammed a group of soldiers.
The soldiers were taking a tour of the city when the Mercedes flat-bed truck drove straight at them. The driver was shot dead at the scene.
MK Avi Dichter, a former head of the Shin Bet, said Sunday that the attack would not affect Israelis' commitment to remain in Jerusalem. Questions were already being raised in the Israeli media why more did not engage the attacker.
The Palestinian Hamas movement, an Islamic militant group that rules Gaza and has killed hundreds of Israelis in attacks over the years, praised the assault but stopped short of taking responsibility.
Islamic radicals have now killed a total of 46 and injured 648 in vehicular attacks in Israel since 2015. Sunday's incident marks the first Israeli deaths in three months.
A number of the victims were trapped under the vehicle, and at least two of the 16 injured were thought to be in a serious condition.
In a statement, Jerusalem's mayor, Nir Barkat, said there was "no limit to the cruelty of the terrorists who use every means to murder Jews and damage the routine life in the capital of Israel".
The Hamas official added, however, that it is not enough and those countries should pressure the Israeli occupiers to implement the UN resolution and leave the Palestinian nation's territories. "The world has received a clear answer from the Palestinians to the peace conference coming up in Paris: More terror", Hotovely said.
The article also reports it is the deadliest attack in months and comes after "a lull in recent violence between Palestinians and Israelis".