The council hopes this will help turn the page on decades of Syrian domination of its neighbour.
Syria however has described the resolution as provocative and biased.
"The insistence of the sponsors of this resolution to pass it ... constitutes uncalled for pressure and provocation that complicates the situation," a statement by the Syrian foreign ministry said on Wednesday.
"Syria cares about the independence and sovereignty of Lebanon and establishing normal Syrian-Lebanese ties independent of outside interference," the statement added.
China and Russia abstained from a 13-0 vote on Wednesday in the 15-member Security Council to implement fully a 2004 measure seeking an end to outside interference in Lebanon, which at the time had been under Syrian domination for 29 years.
The 2004 text, Security Council Resolution 1559, calls for the withdrawal of all foreign forces from Lebanon, free and fair presidential elections, and the disarming of all militias so the government could extend its control to all of its territory.
It has been only partly implemented.
"The insistence of the sponsors of this resolution to pass it ... constitutes uncalled for pressure and provocation that complicates the situation"
Syrian foreign ministry
John Bolton, the US ambassador, welcomed the vote, but acknowledged he would have preferred it to be unanimous.
"It makes clear the burden is now on Syria" to respond to the Lebanese government's request for formal relations and a delineation of the border, Bolton told reporters.
"It clearly says Syria needs to do more to stop the flow of weapons across the Syrian border."
Council diplomats had predicted that Qatar and Argentina would either abstain or vote against the resolution, but both countries voted in favour of the measure drafted by France, the United States and Britain.
Argentina nonetheless reiterated its objection to what it saw as council interference in a question of diplomatic relations, a matter it said should be left to Damascus and Beirut.
Qatar voiced its objection to the drafters' rejection of its request that the resolution also fault Israel for its frequent military flights over Lebanese territory.
Nassir Abdul Aziz al-Nasser, Qatar's UN ambassador, told the council that "these are among the elements hampering the full implementation of Resolution 1559 and are also a breach of Lebanese sovereignty and territorial integrity."