"The count is nearly over and it's a landslide for Saad's list," said the source, who asked not to be identified.
Saad al-Hariri's anti-Syrian bloc had already won nine of the capital's 19 seats in the 128-member parliament before the vote because they were not contested.
The source said candidates on Saad's list had taken all 10 remaining seats.
"This victory is for Rafik al-Hariri. Today Beirut showed its loyalty to Rafik al-Hariri," Saad, 35, told a jubilant crowd celebrating outside his villa in the capital.
"Today is a victory for democracy ... freedom and sovereignty," he said to chanting supporters.
Saad's candidates have taken all
10 remaining Beirut seats
Turnout in the first round of Lebanon's election on Sunday stood at 28%, Aljazeera reported, quoting preliminary official estimates at close of polling.
Disclosing this information at a press conference in Beirut, Lebanon's Interior Minister Hasan al-Sabaa said, "official results would be announced midday on Monday".
According to al-Sabaa, "no security incident that could have affected or obstructed the electoral process took place" during the daylong elections in the capital - the first in a four-stage poll.
"The Lebanese people voted in a democratic atmosphere and in full freedom, without pressure," Aljazeera quoted him as saying.
"We have received congratulations from international observers about the exemplary process of the elections".
More than 400,000 people aged
above 21 were eligible to vote
Earlier, followers of Christian leader Michel Aoun, left off Saad's anti-Syrian ticket, urged people to shun the polls, handing out orange stickers that said: "Boycott the appointments".
The Armenian Tashnag party, disgruntled because the four seats reserved for Beirut's big Armenian community had gone unopposed to Saad's candidates, also demanded a boycott.
"No participation without proper representation for all in Beirut," said Tashnag leaflets in Arabic and Armenian.
The polls follow two political earthquakes in Lebanon – al-Hariri's killing in a bomb blast many Lebanese blamed on Damascus, and the end of Syria's 29-year troop presence.
Between those landmark events, flag-waving Christians and Muslims, including many civil war foes, flooded the streets in protests against Syria, which denied any hand in al-Hariri's death.
For some, Lebanon's first elections in three decades without Syrian troops offer a new start.
"I voted because I believe in change," Basil Eid, 27. "We want Lebanon free of any subordination. We have to rule ourselves by ourselves."
Late Rafik al-Hariri's wife Nazik
cast her ballot in Sunday's vote
For others, the euphoria of the anti-Syrian protests has given way to dismay at politicians who have reverted to electoral bargaining and alliances that curtail voter choice.
"Why should I vote when the result is already decided?" said Abd al-Rahman Itani, in his 40s, near the polling station where the late al-Hariri's wife Nazik cast her ballot.
Armed police and soldiers guarded polling stations in Beirut, where more than 400,000 men and women aged over 21 are eligible to vote. Official results will be declared on Monday.
Only a handful of pro-Syrian leftists and Muslim hardliners are competing with Saad's Future bloc in Beirut.
The solidarity of the anti-Syrian alliance that blossomed after al-Hariri's death has eroded in the run-up to the election.
"Why should I vote when the result is already decided?"
Abd al-Rahman Itani,
Saad's alliance with Druze leader Walid Jumblatt and some Christian foes of Syria is intact, but Aoun, a fierce opponent of Syria just back from exile, was left out in the cold.
Yet, the Saad-Jumblatt front has also made deals with the main pro-Syrian Shia alliance.
Saad's Beirut ticket includes a Hizb Allah candidate. The joint Amal-Hizb Allah list in the south embraces Bahiya al-Hariri, the slain leader's sister.
Tight contests are expected in the north and centre of the country, especially among Christian rivals.