"It has been decided that Bilawal will be the chairman and Mr [Asif Ali] Zardari will be co-chairman," one of the party officials said.
Shortly after the announcement was made, Bilawal told supporters that the party's "struggle for democracy would continue with new vigour".
Both Bilawal and his father Zardari said the PPP would contest the January 8 parliamentary polls.
Zardari also urged Nawaz Sharif, another ex-prime minister and leader of the opposition PML-Nawaz, not to boycott the polls.
Party officials said Bhutto's will had named Asif Zardari as the party chairman. But Zardari requested that the son be named the leader instead and the party leadership agreed.
Al Jazeera's Imran Khan, reporting from Naudero, said that many Pakistanis think that Zardari will actually be leading the party, and Bilawal will perhaps be only a titular leader.
He said: "Many have said that he does not have the stomach to handle politics in this country, because he grew up in the West."
Bilawal - the name means "one without equal" - was born in September 1988, a month before his mother won general elections under military dictator Zia-ul-Haq to become the first female prime minister in the Muslim world.
At Benazir's funeral on Friday, Bilawal was pictured looking composed despite his grief, but analysts say he is perhaps too young to be taking over the job.
"Bilawal is just 19 years old, he needs to be groomed," Talat Masood, political analyst and retired general said.
Sharif's party earlier said it would "review" its boycott of elections if the PPP decides to take part.
Ahsan Iqbal, Sharif's spokesman, said: "If the PPP decides to contest the elections, then we will review our decision whether to contest or continue to boycott."
We announced the boycott after the assassination of Benazir Bhutto to send a message of solidarity and express our sympathy towards her party and the people of her home province. Otherwise, we were going to take part." Murder mystery
Bhutto's death has stoked violence across the country, killing at least 47 people.
Uncertainty has meanwhile intensified over the circumstances of the assassination of Bhutto.
The government claims that Bhutto died from a skull fracture caused by a blow to the head rather than an assassin's bullet.
The PPP has dismissed the government statement, along with claims that she was killed by al-Qaeda, saying the embattled administration of Pervez Musharraf, the Pakistani president, was trying to cover up its failure to protect her.
Sherry Rahman, a PPP spokesperson, said: "This is dangerous nonsense. This is the government absolving itself from taking responsibility for the protection of a former prime minister, one who had been constantly asking for better protection."
Amateur photos, aired on Pakistani TV network Dawn News, have also emerged showing what appears to be a man aiming a gun at Bhutto alongside a white-swathed figure said to be the bomber.
Against such evidence, the government has said the exact cause of her death is not important.
Javed Iqbal Cheema, the interior ministry spokesman, said: "There is no doubt that the fires were shot, there is no doubt that there was a suicide bomber, there was an explosion."
"It doesn't suit the government, you know in no manner and in no way, whether she died of a bullet wound or whether she died of an explosion or because of some other cause."
The government spokesman also said it had no objection to carrying out an official post-mortem, although Bhutto's family rejected an autopsy.
A spokesman for Baitullah Mehsud denied the government's claim that he was behind the assassination, saying: "We don't strike women."
In a recent interview with Al Jazeera in southern Waziristan, Mehsud, a tribal leader alleged to have links to al-Qaeda, said Musharraf is his prime enemy.
Mehsud said: "Our coalition seeks to fight the West in Afghanistan in an organised manner.
However, the barbarism of the Pakistani military and its attacks on Muslims here in the tribal area has put our top priority the expulsion of the military from our areas."
Farhatullah Babar, a spokesman for Bhutto's party, said the government's accusation against Mehsud "appears to us to be a planted story, an incorrect story, because they want to divert the attention".