Foreign Minister Adel al-Jubeir told Fox News the act was a “tremendous mistake” and denied the powerful crown prince had ordered the killing.
He was last seen entering the Saudi consulate in Istanbul on 2 October.
The Saudis, under intense pressure to explain the journalist’s whereabouts, have offered conflicting accounts.
They initially said he had left the building unharmed but on Friday admitted for the first time he was dead, saying that he was killed in a fight. This claim has met widespread scepticism.
Turkish officials believe Mr Khashoggi, a prominent critic of the Saudi government, was murdered by a team of Saudi agents inside the building – and say they have evidence to prove it.
Adel al-Jubeir’s comments on Sunday to Fox News, describing the incident as murder, are some of the most direct to come from a Saudi official.
“We are determined to find out all the facts. And we are determined to punish those who are responsible for this murder,” he said.
“The individuals who did this did this outside the scope of their authority,” he added. “There obviously was a tremendous mistake made, and what compounded the mistake was the attempt to try to cover up.”
He also said that they did not know where the body was, and insisted the action was not ordered by Crown Prince Mohammed bin Salman, seen as Saudi Arabia’s most poweful figure.
“Even the senior leadership of our intelligence service was not aware of this,” he said, calling it a “rogue operation”.
These remarks followed Saudi Arabia’s admission on Friday that Mr Khashoggi had died in the consulate more two weeks earlier. A statement from the public prosecutor said he was killed when a “fist-fight” broke out with some of the people he was meeting inside the consulate.
Until this point – for 18 days – the authorities had maintained that the Saudi critic was last seen leaving the building alive.
But they now said they had arrested 18 people, sacked two aides of Mohammed bin Salman and set up a body, under his leadership, to reform the intelligence agency over the killing.
Both King Salman and the crown prince called Mr Khashoggi’s son on Sunday to express their condolences over his death, the Saudi Press Agency reports.
In a Washington Post interview on Saturday, US President Donald Trump said there had been “deception” and “lies” in Saudi Arabia’s explanation, having previously said he found their narrative to be credible.
He said he would “love” it if the Crown Prince was not responsible for the murder. The president has raised the possibility of imposing sanctions but said halting an arms deal would “hurt us more than it would hurt them”.
The UK, France and Germany issued a joint statement expressing shock at the death and demanding a full explanation, saying: “Nothing can justify this killing and we condemn it in the strongest possible terms.”
German Chancellor Angela Merkel has said she will not allow arms exports to Saudi Arabia to continue given “the current circumstances”, and Canadian PM Justin Trudeau has threatened to cancel a multi-billion dollar defence contract with the kingdom over the affair.
But several of Saudi Arabia’s regional allies have come out in its support.
Kuwait praised King Salman for his handling of the case, while Egypt, Bahrain and the United Arab Emirates have been among those reflecting similar praise
Also on Sunday, Turkey’s president Recep Tayyip Erdogan said he would reveal the “naked truth” of the matter in parliament on Tuesday.
Although Turkey has so far stopped short of officially blaming Saudi Arabia for the killing, investigators have said they have audio and video evidence which shows Mr Khashoggi, was killed by a team of Saudi agents inside the consulate.
Police have searched the nearby Belgrad forest in Istanbul where they believe the body may have been taken and one official was hopeful its fate would be known “before long”.
Both the consulate and the residence of the Saudi consul have been searched too.
Reuters reported on Sunday it had spoken to a Saudi official who said Mr Khashoggi had died in a chokehold after resisting attempts to return him to Saudi Arabia. His body was then rolled in a rug and given to a local “co-operator” to dispose of.
A Saudi operative then reportedly donned Mr Khashoggi’s clothes and left the consulate.
The official said Saudi statements had changed because of “false information reported internally at the time”.
