"We extend an honest call for a national dialogue to come all together in one coalition, under one banner... to reach a compromise, a consensus on which the constitution can be drafted and the new government can be composed," he said.
"There was no loser and winner at all. Whoever is going to win, Libya is the real winner of those elections," Jibril added.
UN leader Ban Ki-moon called for Libya's new leaders to govern in a spirit of "justice and reconciliation" as he hailed the country's first elections since Kadhafi's downfall.
Votes are still being tallied with preliminary results rolled out district by district in a complex counting system that needs to measure the performance of individual candidates as well as those of political entities.
The commission has not given a date for the final results.
NFA secretary general Faisal Krekshi said: "Early reports show that the coalition (NFA) is leading the polls in the majority of constituencies."
If the results are confirmed, Libya, unlike neighbouring Tunisia and Egypt whose strongmen were also ousted in last year's Arab Spring, will buck the trend of electoral success for Islamists.
The leader of the rival Justice and Construction Party, headed by the Muslim Brotherhood's Mohammed Sawan, admitted the NFA had an early lead in the vote count for the capital and Libya's second-largest city of Benghazi in the east.
"But it is a tight race for us in the south," added Sawan, a former political prisoner and member of Libya's Muslim Brotherhood, which launched the party.
The bulk of Libya's population and registered voters are concentrated in Tripoli in the west of the oil-rich desert country and in Benghazi, cradle of the revolt that toppled Kadhafi.
--- 'National interest will triumph' ---
Brahim Saad, secretary general of the National Front for the Salvation of Libya, an Islamist party close to the Muslim Brotherhood, recognised ideological differences would need to be set aside for the sake of national interest.
"We have important deadlines," he said, declining to comment on whether his party was open to joining a coalition led by Jibril before all results roll in.
"We must find common denominators that unite all who participated in the current elections. The question of ideologies will vanish in the General National Congress. National interest will triumph."
Libyans voted for a General National Congress, a 200-member legislative assembly which will steer the country through a transition. Turnout was above 60 percent, the electoral commission said.
Eighty seats in the incoming congress are reserved for political entities while the remainder are held for individual candidates, some of them openly allied to specific parties.
In a country with no history of political parties and no national surveys, forecasting results is virtually impossible, but early media reports seemed to back the party leaders' claims.
Al-Assima TV said the NFA scooped 80 percent of the votes in the district of Tripoli Centre, and delivered a strong performance in the east, with 70 percent support in Benghazi.
"The first winner is the Libyan people," stressed commission chief Nuri Abbar.
Apart from acts of sabotage in the east and one death in Ajdabiya as gunmen opened fire near a polling station, the vote was held in a festive atmosphere in major cities and hailed by international observers as a success.
Ian Martin, head of the UN Mission in Libya (UNSMIL), said on Monday that despite security threats in the east of the country, the elections were an "extraordinary achievement."
Martin added that the dismantling of rival militias will be the key challenge facing the new government.
International monitors and world leaders also said the vote was a huge success for a nation that only months ago was at war and entered the political process with no institutions.
US President Barack Obama set the tone saying: "On behalf of the American people, I extend my congratulations to the people of Libya for another milestone on their extraordinary transition to democracy."
Eastern oil facilities resumed normal operations on Sunday after a three-day protest over the region not being granted a greater share of seats.