Cavusgolu and Pompeo met on the sidelines of a meeting of regional ministers in Singapore.
"At the president's direction, the Department of the Treasury is sanctioning Turkey's minister of justice and minister of interior, both of whom played leading roles in the arrest and detention of Pastor Brunson", she told journalists.
Speaking after the meeting on Friday, Turkish Foreign Minister Mevlut Cavusoglu, said: "Of course you can't expect all issues to be resolved in a single meeting".
Andrew Brunson, who led a Protestant church in the Aegean city of Izmir, was placed under house arrest last week following almost two years in jail on charges of espionage and supporting terror groups.
Asked if the issue threatened Turkey's membership of the North Atlantic Treaty Organization (NATO), Pompeo said: "Turkey is a NATO partner whom the United States has every intention of continuing to work cooperatively with".
Turkey has said the sanctions were unacceptable.
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"The Turks were on notice that the clock had run and it was time for Pastor Brunson to be returned and I hope they'll see this for what it is, a demonstration that we're very serious", Pompeo said.
Also caught in the spat is a plan for the United States military to deliver new F-35 fighter jets to Turkey. "We repeated to them that nothing can be achieved through threatening language and sanctions and we believe that this was well understood", he said.
NASA scientist Serkan Golge, a U.S.
Pastor Brunson was arrested in Izmir, Turkey in October 2016, and was accused without any evidence of helping terrorist groups and of engaging in espionage.
Turkish President Recep Tayyip Erdogan has rejected the USA demands, saying his government won't back down and is willing to "go its own way" if the US acted. Mr. Erdogan's government also alleges the Mr. Brunson has ties to a Turkish preacher now living in exile in the US whom Ankara accuses of fomenting the coup.
The Turkish leader has previously connected Brunson's return to the U.S.to the extradition of Fethullah Gulen, an exiled Turkish cleric who lives in Pennsylvania.
The senior United States general in Europe, Curtis Scaparrotti, was on Thursday holding talks with Turkish military officials in Ankara but there was no indication the Brunson case would be raised. Gulen denies the claim.