Bilawal becomes the third Bhutto to lead the nation's largest political party after his mother and Zulfiqar Ali Bhutto - his grandfather and founder of the PPP - was executed in 1979.
Party officials said Bhutto's will had named Asif Zardari as the party chairman.
But Zardari requested that Bilawal be named the leader, and the party leadership agreed.
Bilawal and his father Zardari said the PPP would contest the January 8 parliamentary polls.
Zardari said: "Despite this dangerous situation, we will go for elections, according to her will and thinking."
He also urged Nawaz Sharif, another ex-prime minister and leader of the opposition PML - Nawaz (PML-N), not to boycott the polls.
The PML-N responded immediately. "We will contest the election," a Sharif spokesman said.
Sharif had earlier urged Zardari to boycott the elections. But he had also said that if PPP decides to contest, he would review his boycott decision.
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 - his name meaning "one without equal" - was born in September 1988, a month before his mother won general elections under Zia-ul-Haq, a military ruler, becoming 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.
Bhutto's death has stoked violence across the country, killing at least 47 people.
As uncertainty intensifies over the circumstances of the assassination of Bhutto, Zardari demanded a UN inquiry into his wife's death, similar to an investigation into the assassination of Rafiq al-Hariri, the ex-prime minister of Lebanon.
He said: "We demand a Hariri commission-style investigation."
The Pakistani government claims that Bhutto died from a skull fracture caused by a blow to the head rather than an assassin's bullet.Murder mystery
The PPP has dismissed the government statement, along with claims that she was killed by al-Qaeda, saying the administration of Pervez Musharraf, the president, was trying to cover up its failure to protect her.
Sherry Rahman, a PPP spokesperson, said: "This is dangerous nonsense. The government [is trying to] absolve 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 local television, 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.
|Zardari, left, is expected to be in|
charge of the party [Reuters]
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 does not suit the government, 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.
The Pakistani government says Baitullah Mehsud, a tribal leader with al-Qaeda links, was the mastermind behind Bhutto's assassination.
Mehsud has denied the charge.
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".