Smoke rose from the camp and loud booms and gunshots could be heard as a helicopter gunship hovered overhead.
Medics said the fighter, who was the fourth Palestinian killed by Israeli troops in the past three days in the West Bank, belonged to the Popular Front for the Liberation of Palestine.
Palestinian ambulances were barred from entering the camp, witnesses said.
The PFLP and Islamic Jihad groups claimed responsibility for killing the soldier in attack using explosives.
The Israelis said he was killed by gunfire.
"Intensive exchanges of fire took place," an army spokesperson said, and added that "violent riots" broke out during which Palestinians threw stones at soldiers.
Troops imposed a curfew in the crowded camp of 5,000 people and searched from house to house, knocking down walls to get from one building to the next, witnesses said.
'Human shield'
Al Jazeera's Shireen Abu Aqla said Israeli forces opened fire at a building in the centre of the camp, saying that there were wanted fighters inside.
Israeli soldiers also prevented Palestinian worshippers from leaving a mosque inside the camp, Abu Aqla reported.
As well as the PFLP, fighters from Izz al-Din al-Qassam Brigades, Hamas' military wing, and al-Aqsa Martyrs Brigades, who are linked to the ruling Fatah faction, battled Israeli forces, Aqla said.
Al-Quds Brigades fighters were also involved in the clashes, she said.
Witnesses said Israeli forces used a resident as a shield as they moved house to house to search of suspected militants.
Israel's supreme court has ruled the practice is illegal and the army says it has respected the ruling.
But Palestinians and human rights groups say that in some cases the army has continued to use civilians as "human shields" during operations in crowded urban areas.