Al-Hariri's coalition takes its name from the date of a rally in 2005 against Syria's military presence in Lebanon following the assassination of Rafiq al-Hariri.
Al Jazeera's James Bays, reporting from Beirut, said the results were not official but all the political parties seemed to be accepting them.
The different parties would now try to reach an agreement on forming a coalition government, he said.
Unity government sought
The Hezbollah alliance, known as the March 8 movement, has not officially conceded, but its main Christian ally, the nationalist Free Patriotic Movement (FPM) led by Michel Aoun, a former army chief, did acknowledged defeat.
He vowed to work with the election winners to form a coalition government.
Hassan Fadlallah, a Hezbollah MP, said Lebanon was ruled by a partnership and the delicate balances or past experiences cannot be changed regardless of the election results.
"What matters to us now is that Lebanon turns a new page, one based on partnership, co-operation and understanding," he told the AFP news agency.
"No party can claim to have won the majority among all communities.
"Whoever wants political stability, the preservation of national unity and the resurrection of Lebanon will find no choice but to accept the principle of consensus."
Michel de Chadarevian, from the FPM, part of the Hezbollah-led alliance, said: "Even if we had won we would have formed a national unity government."
Walid Jumblatt, the Lebanese Druze leader in the March 14 alliance, said Hezbollah and its allies should be included in a new Lebanese government.
"We should not forget that the elections should be a boost to the dialogue and we should not try to isolate the other parties," he said.
|Saad al-Hariri's Future movement leads the
ruling March 14 alliance [Reuters]
Hezbollah has repeatedly called for the formation of a "national unity government" that would give its minority alliance veto power.
The group, which is backed by Syria and Iran, is considered a "terrorist" organisation by the US, but at home is seen by many as a resistance movement that protects Lebanon from Israel.
Security was tight in sensitive areas for fear of clashes between rivals groups.
Security sources said one person was wounded by gunfire in the northern city of Tripoli and there were brawls between rival supporters elsewhere, but there were no reports of serious fighting and voting was relatively trouble-free.
The interior ministry estimated voter turnout in the election had been more than 52.3 per cent, the highest since at least the end of the 1975-91 civil war.