Saudi king hopes Trump brings 'stability' to Middle East
"We wish your excellency success in your mission to achieve security and stability in the Middle East and worldwide," Salman said in a message of congratulations reported by the official Saudi Press Agency.
He praised the relations which are "historic and tight between the two friendly countries, that all parties aspire to develop and reinforce".
The United States and Saudi Arabia have a decades-old relationship based on the exchange of American security for Saudi oil.
But ties between Riyadh and Washington became increasingly frayed during the eight-year Democratic administration of President Barack Obama.
Saudi leaders felt Obama was reluctant to get involved in the civil war in Syria and other regional conflicts.
They also worried that he was tilting towards their regional rival, Shiite Iran, which is on the opposing side to Sunni-dominated Saudi Arabia in the wars in Syria and Yemen.
Washington and other major powers reached an agreement, which took effect in January, to lift international sanctions on Iran in exchange for guarantees that it would not pursue a nuclear weapons capability.