Mr Bashagha’s newly formed government was sworn in at the eastern-based House of Representatives in Tobruk on Thursday, a move that is expected to widen divisions in the country.
Armed groups affiliated with each side have mobilised in the capital in a show of force. Mr Bashagha and Mr Dbeibah both hail from the coastal city of Misurata.
Mr Bashagha’s office also accused his rival of closing off air space and seizing two ministers who tried to travel by land to attend the parliamentary session in Tobruk.
His office said the two ministers were later released.
Mr Dbeibah's government has not responded to those claims.
Meanwhile, Mr Dbeibah's oil minister rejected the National Oil Corporation's explanation that bad weather had halted crude exports and accused the company of undermining security, a sign of the stakes around Libya's oil output of 1.3 million barrels per day.
The UN cast doubt on the validity of the parliament's effort to install Mr Bashagha, saying it was concerned by reports that Tuesday's vote of confidence “fell short of the expected standards".
The Libyan parliament's representative denied there was wrongdoing.
The position of international powers will be key in the coming tussle for control of Libya, with a risk of renewed war after a year and a half of comparative peace between major factions battling for control of the oil-rich country.
Groups located in the main oil-producing regions have issued warnings that they may block off Libya's energy exports.
Libya has had little peace or security since the 2011 Nato-backed uprising that toppled Muammar Qaddafi. In 2014, the country was split between rival administrations — one in the west and the other in the east.
Neither the political nor the military coalitions that are crystallising now exactly match those that were involved in the conflict from 2014 until 2020, when a truce was reached. However, any new conflict is expected to again pit eastern forces against a combination of western groups.
Parliament attempts to take control
Mr Dbeibah's government was put in place a year ago through a UN-backed peace process that was aimed at resolving political problems through an election on December 24. However, the vote did not take place amid arguments over the rules.
Since then, the parliament has tried to take control of the process by saying Mr Dbeibah's term had expired, as it attempts to set a course towards a referendum on an altered constitution and then elections in 2023.
Mr Dbeibah has rejected the parliament's stance and says he is planning to hold national elections in June. Both sides blame each other for the failure of December's election and accuse each other of lacking legitimacy.
The United Nations has not so far recognised Mr Bashagha's legitimacy and raised doubts over the parliamentary process by which his government emerged.
UN Libya adviser Stephanie Williams has pushed for a return to an election process abandoned in December and has invited the parliament and another political body, the High Council of State, for talks.