A second Fatah member died of his wounds hours later while three Palestinian women were wounded by stray bullets.
The atmosphere in the camp was reportedly tense following the shooting, with schools and shops shuttered amid fears partisans from both sides could pursue the conflict.
Ein el-Helweh, 45km south of Beirut, is the largest of Lebanon's 12 refugee camps and home to 47,000 Palestinians or their descendants who fled or were forced from their homes when Israel was created in 1948.