With Prince Mohammed's mediation, Russia on Wednesday released 10 foreigners it had captured in Ukraine, including five Britons and two Americans.
He denied the Crown Prince had become involved to rehabilitate his reputation.
U.S. citizens Alexander Drueke, 39, and Andy Huynh, 27, both from Alabama, are expected to leave Saudi Arabia within days, officials said.
But Saudi Arabia has shown little readiness to join the effort to isolate Russia.
U.S. National Security Adviser Jake Sullivan, Secretary of State Antony Blinken and British Prime Minister Liz Truss thanked the Saudi crown prince for his role.