The vote paves the way for Hariri's anticipated visit to Syria - a major supporter of Hezbollah - for talks with the country's president.
No firm date has yet been set for the visit.
Hariri's coalition, backed by the United States and Saudi Arabia, had accused Syria of assassinating Rafiq al-Hariri, Saad's father, in February 2005.
Syria denies any involvement, but the killing forced it to end its 29-year military presence in Lebanon in April 2005.
A special court was set up in The Hague to investigate and prosecute the killers.
Christian MPs and ministers of the Hariri-led parliamentary majority voiced discontent over the clause, which states the right of "Lebanon, its government, its people, its army and its resistance" to liberate all Lebanese territory.
But they backed the cabinet in the vote on Thursday.
Before the vote, Hariri told the assembly that Lebanon was looking for better relations with Arab countries.
"We specially look forward to Lebanese and Syrian relations founded on brotherly ties ... based on respecting the sovereignty of both countries," he said.
Lebanon remained without a functioning government for six months after June elections.
Weapons not discussed
But Syria and Saudi Arabia, keen to improve bilateral ties and ease political and sectarian tensions in Lebanon, have nudged their allies to agree on a unity government that gave Hariri's coalition 15 ministers in a 30-member cabinet.
Hezbollah and its allies got 10 and Michel Suleiman, the Lebanese president, five.
Hezbollah, which fought Israel in 2006, regularly states its weapons are not open to discussion.
It argues its arms are necessary to protect the country against any future aggression by Israel, which withdrew from southern Lebanon in 2000 after a 22-year occupation.
Hariri's US and Western-backed alliance defeated a Hezbollah-led opposition supported by Syria and Iran in the June general elections.