Fayez al-Sarraj's arrival at a naval base yesterday drew fury from the militia-backed authority in charge of Tripoli, which demanded he leave or surrender.
Gunmen stormed the headquarters of a Libyan television station overnight, apparently in support of the new government, but the capital appeared calm today.
Banks and shops were open, police were posted on the streets and flights had resumed at Metiga airport after being suspended the day before "for security reasons".
The international community hailed the new government's arrival as a crucial step in restoring order to Libya, which has been wracked by chaos since the 2011 overthrow of Moamer Kadhafi.
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Formed under a power-sharing deal agreed in December, the unity government is meant to take over from rival groups running the country.
Libya has had two administrations since mid-2014 when the militia alliance overran Tripoli, setting up its own authority and forcing the internationally recognised parliament to flee to the country's remote east.
The United States and its European allies have threatened action against those who undermine the political process.
EU member states today agreed to impose sanctions on three Libyans for obstructing the formation of Sarraj's government.
One European source said the measures comprise "a ban on travelling in the European Union and a freeze on assets in the EU."
A European diplomatic source told AFP recently that EU sanctions would target the Tripoli government's prime minister Khalifa Ghweil, the head of the General National Congress Nuri Abu Sahmein, and Aguila Saleh, speaker of Libya's internationally recognised government.