The call was made at a ministerial meeting hosted by Tunisia and attended by delegates from Algeria, Chad, Egypt, Niger and Sudan, as well as the UN envoy to Libya.
Libya's neighbours said they fully backed the unity government and stressed the "necessity to speed up its departure for Tripoli," a statement said.
Libya has had two rival administrations since mid-2014 when the recognised government was forced from Tripoli to the far east after a militia alliance including Islamists overran the capital.
The United Nations is pushing the two sides to accept a unity government, created under a power-sharing deal agreed by the rival parties in December.
But it failed to obtain formal parliamentary approval from both administrations, a move effectively blocking its declared will to take office and move to the capital.
The jihadist Islamic State group has taken advantage of the chaos to spread its influence in Libya claiming devastating attacks in the North African nation and in its neighbours, namely Tunisia.
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At the onset of the meeting, Tunisian Foreign Minister Khemaies Jhinaou warned that "the proliferation of terrorist groups and their control of certain regions in Libya is a source of extreme concern... a danger for Libya's people.. and for the stability of its neighbours."
UN envoy Martin Kobler exhorted Libya's neighbours to back efforts by the United Nations to install a unity government in the oil-rich country.
"The (political) process remains precarious... At the same time, the terrorists are taking advantage of the political divisions and the Libyans, as well as their neighbours, continue to bear the consequences," said Kobler.
"Daesh (an Arabic acronym for IS) in Libya is a growing and imminent threat," he added.
The UN envoy also called for the formation of a united Libyan army that would include General Khalifa Haftar.
"He must be part of a solution," said Kobler.
Haftar heads the armed forces loyal to the internationally recognised government and his ouster is demanded by Islamist-backed administration in Tripoli.
The general returned to Libya after more than 20 years in exile in the United States to join the 2011 uprising that toppled dictator Moamer Kadhafi. He has since vowed to crush Islamists.