The fighting between the Jund al-Sham extremist group and members of Palestinian President Mahmud Abbas's Fatah movement prompted hundreds of residents to flee Ain al-Helweh camp and shelter in nearby mosques.
Medical sources said at least 35 people were wounded, with ambulances unable to enter the camp to retrieve more injured because of the intensity of the clashes.
At least two of the dead were Fatah members, one of them an officer, Palestinian sources said. The identity of the third person was unclear.
The sound of fierce gunfire and rocket fire could be heard in neighbouring Sidon on Tuesday morning, although the fighting appeared to have eased somewhat by midday.
The Lebanese army reinforced its positions at the four main entrances to Ain al-Helweh.
Soldiers were allowing those able to reach the entrances to leave the camp, but preventing anyone from entering, an AFP correspondent said.
At a mosque near the camp, some 900 people were sheltering after fleeing the violence, many of them Palestinians already displaced by the war in Syria.
The courtyard was filled with children playing games, while adults mostly sat inside.
One lady prepared sandwiches as others listened to the news on the radio to try to find out how long they might be displaced for.
A man who gave his name as Abu Khaled, originally from the Yarmuk refugee camp on the outskirts of Damascus, bemoaned the cramped conditions in the overwhelmed mosque.
"There are charities giving out bits of aid, but it isn't going to everyone," he said.
"I regret leaving Syria. I could have taken refuge in Damascus instead of in a war zone like Ain al-Helweh," he added.
Jamila Shami, who sat hugging her three children in the mosque's courtyard, said they had fled when they heard the sound of fighting.
"We left the camp so quickly we had no time to bring luggage or food with us," she said.
"We cannot bear any more suffering. Our children ask us why we didn't leave Yarmuk for somewhere inside Syria.
"In Yarmuk we used to hear bullets and shelling and here we hear it too."
The clashes in Ain al-Helweh first broke out on Saturday after two Fatah members were killed during an apparent assassination attempt by Islamists on a leading Fatah official.
They continued sporadically throughout the weekend.
Tensions have been running high for months between Palestinian factions and Islamists inside Ain al-Helweh, which is impoverished and overcrowded.
By long-standing convention, the army does not enter Palestinian refugee camps in Lebanon, leaving the factions themselves to handle security.
That has created lawless areas in many camps, and Ain al-Helweh has gained notoriety as a refuge for extremists and fugitives.
But the camp is also home to more than 54,000 registered Palestinian refugees who have been joined in recent years by thousands of Palestinians fleeing the conflict in Syria.
More than 450,000 Palestinians are registered in Lebanon with the UN agency for Palestinian refugees UNRWA.
Most live in squalid conditions in 12 official refugee camps and face a variety of legal restrictions, including on their employment.
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