The draft, which was submitted by Britain, had been the subject of tough negotiations among the 15 council members, and was amended several times before the vote.
It also endorses the results of UN-brokered peace negotiations in Sweden last week. Yemen's warring parties agreed to a ceasefire that took effect on Tuesday and the withdrawal of fighters in Hodeida, a major gateway for aid and food imports.
UN Security Council to vote on sending observers to Yemen
The city is a vital lifeline for millions at risk of starvation, and the ceasefire between Saudi-backed government forces and Huthis is seen as the best chance yet of ending four years of devastating conflict.
The agreement also included a planned prisoner swap involving about 15,000 detainees.
The UN Security Council resolution "insists on the full respect by all parties of the ceasefire agreed" for Hodeida.
It authorises the United Nations to "establish and deploy, for an initial period of 30 days from the adoption of this resolution, an advance team to begin monitoring" the ceasefire, under the leadership of retired Dutch general Patrick Cammaert.
Saudi Arabia welcomed the resolution which means the Huthis "will lose their margin of maneuver," Khalid Manzlawi, the kingdom's deputy permanent representative to the UN, said in a statement carried by the official Saudi Press Agency.
He also thanked Kuwait and Saudi Arabia's ally the United States "for reaching the appropriate formula for the resolution, which is in the interest of the people of Yemen and the maintenance of international peace and security."
According to the UN, Cammaert - who served multiple times as a UN peacekeeper - was expected in the Jordanian capital Amman before heading to the militant-held capital Sanaa and Hodeida.
The resolution also authorises UN Secretary General Antonio Guterres to "submit proposals as soon as possible before December 31, 2018 on how the United Nations will fully support the Stockholm Agreement as requested by the parties."
Saudi ambassador to Yemen says UN agreement requires Houthi withdrawal from Hodeidah
French ambassador Francois Delattre said the unanimous vote sent a "strong signal of the council's unity and engagement" on Yemen, and that it had put its weight behind the UN-brokered talks.
Diplomats said the UN observer mission could consist of 30 to 40 people, tasked with ensuring the withdrawal of the warring parties from Hodeida and the safe passage of humanitarian aid.
The UN said the first members of the mission were already en route to the region.
The observers will head up monitoring teams made up of government and Huthi representatives, under the auspices of a Redeployment Coordination Committee headed by Cammaert.
A week ago, following the talks in Sweden, UN envoy Martin Griffiths asked the Security Council for the quick deployment of observers.
But the negotiations were particularly tough, with Russia at one point threatening to use its veto if a mention of Iran supporting Huthi rebel attacks - inserted by the United States - was not removed.
In a compromise, the phrase "further condemning the supply, from Iran and other actors" of the Huthis became "the supply, from whatever source."
But in the end, the entire paragraph was scrapped from the final version.
The war in Yemen between the rebels and troops loyal to President Abedrabbo Mansour Hadi escalated in 2015, when he fled into Saudi exile and the Saudi-led military coalition intervened.
Since then, the conflict has killed about 10,000 people, according to the World Health Organisation, although human rights groups say the real death toll could be five times as high.
Tales of displaced Yemenis: two camps, same tragedy
On December 8, the UN said that as many as 20 million people in Yemen were "food insecure," calling the situation the "world's worst humanitarian crisis."
Louis Charbonneau, the UN director for Human Rights Watch, said the resolution "sends an important message to the suffering people of Yemen that they haven't been forgotten."
Charbonneau also called on the Security Council to consider imposing "targeted sanctions" on those who violated the laws of war in Yemen, including "senior Saudi, Emirati and Huthi officials."
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