Journalist Jamal Khashoggi was killed in a fight in the Saudi consulate in Istanbul, the country's state TV reported quoting an initial probe.
It said deputy intelligence chief Ahmad al-Assiri and Saud al-Qahtani, senior aide to Crown Prince Mohammed Bin Salman, were dismissed over the affair.
US President Donald Trump said what happened was "unacceptable" but Saudi Arabia was a "great ally".
This is the first time the kingdom has admitted Mr Khashoggi has died.
Speaking at a round table event, President Trump said the arrests were an important "first step". He praised the kingdom for acting quickly, and while he said sanctions were an option against the country, he spoke of the possible effect such moves would have on the US economy.
Asked if he found Saudi Arabia's version of events credible, he replied, "I do."
Saudi King Salman has also reportedly ordered the formation of a ministerial committee, headed by Crown Prince Mohammed, to restructure the intelligence services.
The journalist was last seen entering the Saudi consulate in Istanbul on 2 October, to pick up paperwork that would allow him to marry his fiancée Hatice Cengiz.
Reports on Saudi state media followed shortly after King Salman spoke on the phone to Turkey's President Recep Tayyip Erdogan about the case.
Saudi Arabia reportedly acted on information provided by Turkish authorities as part of its inquiry, investigating a number of suspects.
Earlier police in Turkey expanded their search for Mr Khashoggi's body, with unnamed officials saying his body may have been disposed of in the nearby Belgrad forest or on farmland.
What are Saudi officials saying?
A statement from Saudi Arabia's public prosecutor said a fight broke out between Mr Khashoggi and people who met him in the consulate - ending with his death.
The investigations are still under way, it said, and 18 Saudi nationals have been arrested.
Media reports also spoke of the dismissal of the two senior officials.
Saud al-Qahtani is a prominent member of the Saudi Royal Court and adviser to Prince Mohammed bin Salman.
Major-General Ahmed al-Assiri has acted as the top spokesman for the kingdom about the war in Yemen.
Gen Assiri spoke to the BBC in 2017 about the conflict, defending Saudi Arabia's actions.
What's been the reaction?
A White House statement acknowledged the Saudi investigation shortly after the announcement.
It said the US was "deeply saddened" to hear confirmation of Mr Khashoggi's death.
President Trump stressed the importance of Saudi Arabia as a counterbalance to Iran in the Middle East, and pushed back against the need for sanctions against the country in light of the new information, talking about the effect of such a move on the US economy.
He spoke of his visit to Saudi Arabia - his first trip abroad as president - and the $110bn (£84bn) arms deal he signed with the kingdom.
"I'd rather keep the million jobs [in the US] and find another solution," he said.
Earlier this week Mr Trump said there would be "very severe" consequences if Saudi Arabia was proved to have killed the journalist.
US Senator Lindsey Graham, a Republican highly critical of the Saudis, said he was "sceptical" of the report on the journalist's death.
Saudi Arabia changes its story
By James Landale, BBC Diplomatic Correspondent
Saudi Arabia has finally changed its tune.
After claiming for days that Jamal Khashoggi had left the country's consulate in Istanbul on 2 October, it has now admitted that he died there.
But the Saudi description of how the journalist died is in stark contrast to the account given by Turkish security sources to local newspapers. While they suggested he was deliberately tortured and dismembered with a bone saw, the Saudi version - 17 days on - claims Mr Khashoggi died after a fight broke out.
In other words, the Saudis are suggesting his death was not premeditated.
This is the "botched rendition" explanation - a kidnap attempt that went wrong after which the killers attempted a gruesome cover-up.
The question now is whether Riyadh's Western allies will find this account convincing and persuade them not to take punitive action against Saudi Arabia.
I think we can safely predict some scepticism. What might impress countries like Britain and the United States more is the dismissal of two key advisers to the Saudi crown prince, the country's de facto leader.
One Western diplomat told me they were "not just part of [Prince Mohammed]'s inner circle. They were his inner circle". Their sacking will be seen as an attempt to ring fence the crown prince from accusations that he knew about the killing.
The question now is whether this initial line of defence will hold. Some Western diplomats are expecting - or perhaps hoping - that the crown prince's wings may be clipped even more, perhaps with another royal prince named as deputy crown prince with an alternative power base.
What happened to Jamal Khashoggi?
Mr Khashoggi - a prominent journalist who fell out of favour with the Saudi government - had been living in self-imposed exile in the US since last year.
He was last seen entering the Saudi consulate in Istanbul on 2 October, to pick up paperwork that would allow him to marry his fiancée Hatice Cengiz.
Turkish officials believe Mr Khashoggi was killed by a team of Saudi agents inside the consulate, and his body then removed.
Saudi Arabia has denied the claims, and initially insisted Mr Khashoggi had freely left the embassy.
Why does Turkey say he was murdered?
Turkish officials say they have audio and video recordings that show Mr Khashoggi being murdered.
Turkish newspapers with close links to the government have published gruesome details of the alleged audio, including what they describe as the sounds of screams and Mr Khashoggi being interrogated and tortured.
Meanwhile, Turkish media say they have identified a 15-member team of suspected Saudi agents who flew into and out of Istanbul on the day of the disappearance.
Jamal Khashoggi disappearance: 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: Mr 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
- Turkish government announces Mr Khashoggi is missing, thought to be in the consulate
- Saudi Arabia says he left the embassy
- Turkish officials tell the BBC they believed Mr Khashoggi was killed at the consulate. This is later strongly denied by Saudi Arabia
- 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
- 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