The group is known as the "March 8" alliance.
The US, which lists Hezbollah as a "terrorist" group, has linked future aid to Lebanon to the shape and policies of the government that replaces the current national unity cabinet.
A win for Hezbollah and its Maronite Christian allies could precipitate a tilt towards Syria.
Nasrallah Boutros Sfeir, the influential Christian Maronite patriarch, gave warning on Saturday of "a threat to the Lebanese entity and its Arab identity".
His remarks have been seen as a veiled attack on Hezbollah, which enjoys backing from Iran.
By contrast, the governing "March 14" coalition is backed by the US and other countries, including Saudi Arabia.
It swept to power in 2005, amid a wave of popular discontent following the murder of Rafiq al-Hariri, a former prime minister, in a massive car bombing in Beirut, and is led by Saad al-Hariri, Rafiq al-Hariri's son.
Israel has said been clear that, should Hezbollah win, it will treat Lebanon as an enemy state.
But Jimmy Carter, the former US president, who is acting as an election observer in the Lebanese vote, suggested the US administration could engage with the Lebanese government led by Hezbollah.
"I was talking last night to the White House representatives," he told reporters.
"They reminded me that in [US] President [Barack] Obama's speech he made it very clear that the United Sates would accept the results of elections around the world from a democratic society, if they are honest and fair and free."
Conducted under a new electoral law, this year's vote will largely be decided by Lebanon's divided Christian districts.
Aoun faces Christian rivals in the shape of the Phalange party led by Amin Gemayel, a former president, and the Lebanese Forces led by Samir Geagea, another March 14 camp leader.
Al Jazeera's James Bays, reporting from Lebanon, said the Christians looked set to emerge as the election "kingmakers".
"To put it simply, the Christians are bitterly divided. The Christians find themselves in the two different camps - there are Christian parties on both sides of the political divide," he said.
"They are the fluid voters that might go either way. So the Christians are going to end up - even though they're very divided - as the kingmakers."
In the run-up to the elections, much of the campaigning has focused on Hezbollah's military power, which is stronger than the state's security forces.
|The build up to Lebanon's elections
has largely been calm[EPA]
Opponents say Hezbollah's weapons undermine the state, while the group and its allies see them as crucial to defending Lebanon from Israel.
Sectarian tension brought Lebanon to the brink of civil war last year when more than 100 people were killed in violence before an agreement led to the election of Michel Sleiman, then the army chief, as president and the formation of a national unity government.
Lebanon's power-sharing system divides the 128 seats in parliament equally between Christians and Muslims.
The country's president must be a Maronite Christian, the prime minister a Sunni Muslim and the speaker of parliament a Shia Muslim.
Polls are due to close at 7pm local time (16:00 GMT).