"We call on all Libyan public institutions to facilitate a peaceful and orderly handover of power so that Libya’s new leaders can begin to govern from Libya's capital," US Secretary of State John Kerry said in Paris after a meeting with his counterparts from France, Britain, Italy and the European Union.
In a joint statement released after the meeting, the allies said the UN-backed unity government should move to Tripoli as soon as possible.
The presidential council, headed by Fayez al-Sarraj, announced Saturday it was taking power despite lacking parliamentary approval.
Libya has had two rival administrations since mid-2014 when the recognised government was forced from Tripoli after a militia alliance including Islamists overran the capital.
A UN-brokered agreement in December between representatives of the rival parliaments provided for a power-sharing government to be based in Tripoli.
However last month, 100 lawmakers from the internationally recognised parliament said they supported a UN-backed unity government but were "forcibly prevented" from putting a new reduced cabinet line-up to a vote of confidence.
Al-Sarraj said in a statement on Saturday that the majority petition signed by lawmakers was equivalent to a vote of confidence.
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He urged institutions "to immediately make contact with the unity government in order to organise the modalities of passing over power in a peaceful and organised way".
In Paris, the foreign ministers and European Union top diplomat Federica Mogherini warned that any individuals who "undermine the political process" will face sanctions.
"We are working with the EU and with the United States to quickly adopt sanctions if it proves necessary, against those in both camps who prevent the new government from taking power," said French Foreign Minister Jean-Marc Ayrault.
The EU's 28 foreign ministers will discuss the sanctions tomorrow, which could include the freezing of assets and a travel ban within Europe.
They are likely to target the speaker of Libya's internationally recognised parliament, Aguila Saleh, as well as Nuri Abu Sahmein of the Tripoli-based General National Congress and its head Khalifa Ghweil.
Libya descended into chaos after the 2011 ouster of longtime dictator Moamer Kadhafi, allowing extremist organisations including the Islamic State group to gain significant ground.
Western countries have agreed that military action is needed to dislodge IS from Europe's doorstep, but world powers want a national unity government installed to request help before formally intervening.
"Political unity and an inclusive and functioning government is the only way to put an end to the instability that has fuelled the development of terrorism in Libya," read the joint statement from the US and its EU allies.