Two Hamas officials said earlier four people had been killed. The statement issued by Hamas said members of the National Security Forces, who belong to the Palestinian Authority, carried out the attack.
Palestinian Authority officials condemned the attack, rejected Hamas' allegation and urged all sides to wait for the results of an investigation.
The Palestinian Authority and Hamas have been bitter rivals since the 2007 brief civil war in the Gaza Strip, which ended up in Hamas's control, while the Palestinian Authority continued to have limited control of the Israeli-occupied West Bank. Several reconciliation attempts have since failed to end the power-sharing disputes between the two sides.
The shootings took place during the funeral of a Hamas supporter who was killed in an explosion on Friday night in the camp in the southern Lebanese port city of Tyre.
"We hold the Authority's leadership in Ramallah and their security services in Lebanon fully responsible for the crime," the Hamas statement said.
The Lebanese army said one suspect in the shooting has been taken into custody by the Palestinian National Security.
Speaking by phone from Beirut, the Palestinian ambassador to Lebanon, Ashraf Dabour, rejected Hamas allegations.
"This is a rejected and a condemned action... Investigation committees will reveal who stood behind it," Dabour said. "We have made contacts with Hamas leaders and demanded they wait for the investigation results."
Hamas said in a statement on Saturday that the blast on Friday night was caused by an electrical fault in a warehouse containing oxygen and gas cylinders for COVID-19 patients, as well as detergents and disinfectants.
A number of armed Palestinian factions, including Hamas and the Fatah movement, hold effective control over roughly a dozen Palestinian camps in the country, which Lebanese authorities by custom do not enter.