Jubeir's comments come the day after Turkish President Recep Tayyip Erdogan called for the extradition of 18 Saudi nationals authorities say were involved in the murder of Khashoggi, a Saudi government critic killed in his country's consulate in Istanbul this month.
Saudi officials also admitted this week that those agents plans to murder Khashoggi - who has been critical of the Saudi kingdom - was "premeditated".
The kingdom initially said it knew nothing about what happened to Mr Khashoggi, who was a vocal critic of Saudi Crown Prince Mohammed bin Salman, but then gradually changed its position.
Saudi officials have also said he was accidentally killed in a botched security operation to return him to the kingdom.
It also came after Saudi Arabia's Crown Prince Muhammad bin Salman on October 24 called the murder "repulsive" and vowed that justice will prevail, in his first public comments on a case that has provoked an global outcry. "Failure of any one nation to adhere to worldwide norms and the rule of law undermines regional stability at a time when it is needed most".
"We will prove to the world that the two governments (Saudi and Turkish) are cooperating to punish any criminal, any culprit, and at the end justice will prevail", he said to applause.
The fiancee of slain Saudi journalist Jamal Khashoggi has said she did not accept an invitation from US President Donald Trump to visit the White House because she thought it was aimed at influencing public opinion in his favour.
However, world leaders including Prime Minister Theresa May and President Trump have called the Saudi's account "not credible" with the United States leader saying it was the "worst cover up in the history of cover ups".
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German Chancellor Angela Merkel vowed on Friday that Berlin would not export arms to Saudi Arabia until the murder is clarified, a stance French President Emmanuel Macron dubbed "pure demagoguery" as it "has nothing to do with Mr Khashoggi".
He pressed Riyadh to reveal who ordered the killing and the whereabouts of the body.
In an interview with Turkish broadcaster Haberturk that aired Friday, Cengiz said Khashoggi was treated well during an earlier trip to the consulate on September 28.
"We will overcome" the consequences of the Khashoggi killing, he added. Turkish media have published details of his alleged torture and decapitation.
"His local network in Turkey was very good as you know, his political network as well", Cengiz said.
And he made clear that the USA commitment to the region outpaces any presence by Russian Federation, which he said lacks essential moral principles.
The State Department declined to say Friday what further actions the administration may take, but an official told ABC News, "The Secretary continues to engage with Saudi leaders at the highest levels".
Referring to Trump's invitation, she said: "I perceived it as a statement to win public favour".