Foreign Minister Adel al-Jubeir told Fox News "the murder" had been a "tremendous mistake" and denied the powerful crown prince had ordered it.
The journalist was last seen entering the Saudi consulate in Istanbul.
The Saudis, under intense pressure to explain Khashoggi's whereabouts, have offered conflicting accounts.
They initially said he had left the building unharmed on 2 October but on Friday admitted for the first time he was dead, saying he had been killed in a fight. This claim met widespread scepticism.
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Turkish officials believe 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.
How has the Saudi version of events changed?
Adel al-Jubeir's comments, 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 had not been ordered by Crown Prince Mohammed bin Salman, seen as Saudi Arabia's most powerful figure.
"Even the senior leadership of our intelligence service was not aware of this," he said, calling it a "rogue operation".
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However, Yeni Safak, a media outlet close to Turkey's government, says it has information showing that the office of the crown prince had received four phone calls from the consulate after the killing.
The paper, which has leaked many details of the Turkish investigation so far, suggested embassy official Maher Mutreb had used his own mobile phone to call the office, as well as an American number believed to belong to the crown prince's younger brother, Khaled, who was Saudi ambassador to the US.
Prince Khaled bin Salman left the US soon after Khashoggi's disappearance.
Saudi Arabia admitted for the first time on Friday that Khashoggi had died, suggesting he had been killed in a fight 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.
Jamal Khashoggi killing: The key events
03:28: A private jet carrying suspected Saudi agents arrives at Istanbul airport. A second joins it late afternoon
12:13: Several diplomatic vehicles are filmed arriving at the consulate, allegedly carrying some of the Saudi agents
13:14: Khashoggi enters the building, where he is due to pick up paperwork ahead of his marriage
15:08: Vehicles leave the consulate and are filmed arriving at the nearby Saudi consul's residence
21:00: Both jets leave Turkey by 21:00
3 October - Turkish government announces Khashoggi is missing, thought to be in the consulate
4 October - Saudi Arabia says he left the embassy
7 October - Turkish officials tell the BBC they believe Khashoggi was killed at the consulate. This is later strongly denied by Saudi Arabia
13 October - Turkish officials tell BBC Arabic they have audio and video evidence of the killing. The existence of such tapes had previously been reported by local media
15 and 17-18 October - Forensic teams carry out searches of consulate
19 October - Saudi state TV reports an initial investigation shows Jamal Khashoggi died in the consulate. Two Saudi senior officials are dismissed and King Salman announces the formation of a ministerial committee to restructure the intelligence services
20 October - Turkish officials vow to reveal all evidence relating to the killing
21 October -Saudi foreign minister tells Fox News it was a "rogue operation" and denies the crown prince ordered the "murder"
What are they doing about it?
They say they have 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 Khashoggi's son, Salah, on Sunday to express their condolences over his death, the Saudi Press Agency reports.
Salah Khashoggi resides in Saudi Arabia and, according to the Wall Street Journal, had been barred from leaving the country to visit his father who was living in self-imposed exile in the US.
Meanwhile, Jamal Khashoggi's fiancée. Hatice Cengiz, who raised the alarm about his disappearance after waiting for him for hours outside the consulate, was given 24-hour police protection, Turkish state news agency Anadolu reports.
How has the world reacted?
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 Prime Minister Justin Trudeau has threatened to cancel a multi-billion dollar defence contract with the kingdom over the affair.
Yasin Aktay, an adviser to Turkish President Recep Tayyip Erdogan, dismissed the Saudi version of events, writing in Yeni Safak newspaper: "One cannot help but wonder how there could have been a 'fistfight' between 15 young expert fighters... and a 60-year-old Khashoggi, alone and defenceless."
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.
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President Erdogan said he would reveal the "naked truth" of the matter in parliament on Tuesday.
Where is the investigation now?
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 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.
Turkish prosecutors have also been taking statements from the consulate's locally employed staff.
Reuters news agency reported on Sunday it had spoken to a Saudi official who said 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 Khashoggi's clothes and left the consulate.
The official said Saudi statements had changed because of "false information reported internally at the time."