The draft resolution, written by France and approved unanimously, would allow for the approximately 10,000 members of the United Nations Interim Force in Lebanon (UNIFIL) - which was created in March 1978 - to stay in place.
According to the draft text, the Security Council warned that "violations of the cessation of hostilities could lead to a new conflict that none of the parties or the region can afford."
Lebanon says Israel drone attack is a 'declaration of war'
It "condemns all violations of the Blue Line" between Lebanon and Israel, "both by air and ground, and strongly calls upon all parties to respect the cessation of hostilities."
The Lebanese army fired on an Israeli drone in the southern part of the country on Wednesday.
That incident came just days after sectarian movement Hezbollah blamed an armed drone attack in its south Beirut stronghold on Israel - an incident that President Michel Aoun said amounted to a "declaration of war."
"Urging all parties to make every effort to ensure that the cessation of hostilities is sustained," the Security Council called on all sides to "exercise maximum calm and restraint."
The resolution includes a requirement - on the insistence of the United States, diplomats said - for the UN secretary-general to perform an evaluation on the UNIFIL mission and its staff before June 1, 2020.
Hezbollah chief threatens Israel after Beirut 'drone attack'
Washington was unable, however, to reduce the maximum allowed number of peacekeepers deployed to 9,000.
Also at the US's request, the Security Council resolution calls for UNIFIL to have full access to the Blue Line.
It expresses "concern that UNIFIL still has not been able to access all relevant locations north of the Blue Line related to the discovery of tunnels crossing the Blue Line."
Israel has said it discovered and destroyed a series of cross-border Hezbollah tunnels since late last year.
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