Eighteen Saudis are arrested and the state news agency says the kingdom "expresses its deep regret" over Mr Khashoggi's death.
Mr Khashoggi, 59, went missing on 2 October during a visit to the consulate to get marriage papers and pressure had been growing on Saudi Arabia to explain his disappearance.
Saudi Arabia's public prosecutor was quoted on state television as saying that a fight broke out between Mr Khashoggi and people who met him in the consulate, leading to his death.
An official Saudi source said "discussions" between Mr Khashoggi and others at the consulate "did not go as required and developed in a negative way, leading to a fight and a quarrel".
The source added that "the brawl aggravated to lead to his death and their attempt to conceal and cover what happened".
Eighteen Saudis have been arrested so far in connection with the journalist's death and the state news agency said the kingdom "expresses its deep regret" over Mr Khashoggi's death.
In other developments, one of Saudi Arabia's top intelligence officials, Ahmed El Asiri, has been sacked, as has royal court adviser Saud Al Qahtani.
The Saudi rulers were expected to say that General Asiri received orders from Prince Mohammed bin Salman to capture Mr Khashoggi for an interrogation in Saudi Arabia, but either misunderstood or overstepped them, according to two of the people familiar with Saudi plans who were quoted in the New York Times.
An official, speaking to Reuters news agency, said at the time: "We believe that the murder was premeditated and the body was subsequently moved out of the consulate."
The prominent journalist has interviewed Osama bin Laden and likened the Saudi Crown Prince to Vladimir Putin.
A senior police source told online news website Middle East Eye the journalist had been "brutally murdered, killed and cut into pieces".
"Everything was videotaped to prove the mission had been accomplished and the tape was taken out of the country," the source said.
US president Donald Trump was criticised for not being tough enough on Saudi Arabia, saying that it was "too early" to determine consequences.
Turkish officials accused Saudi Arabia of a state-sponsored killing and a number of countries, including the UK, pulled out of a major investment summit in Riyadh next week in protest.
Sky News special correspondent Alex Crawford said from Istanbul that confirmation of Mr Khashoggi's death was "not too much of a surprise from our sources in Turkey".
"I think the Saudi just thought there was nowhere else to go - they had to admit it.
"Now they've been backed into a corner where they've had to admit something very badly went wrong."
Mr Khashoggi, a Washington Post columnist, fled Saudi Arabia for Washington in September 2017, months after Prince Mohammed was appointed heir to the throne.
He had been criticised by Saudi authorities for being too progressive and he had described Prince Mohammed as a "brash and abrasive young innovator" - and even said he was "acting like Putin".