Walid Jumblatt, the Druze politician and head of the Progressive Socialist Party, also endorsed the bill.
"The Palestinian refugees have been waiting [for] 62 years, and we are not granting them the minimum of their rights," he said.
A group of Christian parties, including the Phalange party and Lebanese Forces, opposed the measure, warning that it was a first step towards naturalising Lebanon's refugee population.
They worry that Palestinian refugees might eventually become Lebanese citizens. Most are Sunni Muslims, so their naturalisation would tip Lebanon's sectarian balance.
Sami Gemayel, a member of the Phalange party, said it would be unconstitutional.
The constitution includes a provision barring the "settlement of non-Lebanese in Lebanon".
"The Palestinian matter is a disputed issue, and a source of fear," Elie Marouni, an MP from the Phalange party, said.
Jumblatt called that position "stupid" and criticised "right-wing parties" for failing to distinguish between granting rights to Palestinian refugees and offering them citizenship.
The refugees are planning several major protests across Lebanon on June 27 to demand civil rights.
More than 400,000 Palestinian refugees live in camps scattered across Lebanon. They do not enjoy the same rights as Lebanese citizens: refugees are barred from owning property, working, travelling overseas and receiving basic social services from the government.
The crowded camps have given rise to several armed groups. The Lebanese army fought a three-month battle in 2007 with one group, Fatah al-Islam, at the Nahr al-Bared refugee camp near Tripoli. More than 400 people, including dozens of civilians, were killed in the fighting.
Hariri said the poor conditions in the camps helped to create such groups.
"Should we deprive Palestinian refugees in Palestinian camps of their rights, so that they become terrorists in the future?" Hariri said during the parliamentary debate. "We have a historic opportunity to vote on this proposal."
But the proposal is controversial within Hariri's Future bloc too.
"Do not give to them [Palestinian refugees] the rights which are given to the Lebanese people," Atef Majdanani, a Future Movement MP, said.