MOSCOW—The first visit by a Saudi monarch to Moscow showcased Russia’s efforts to wield its rising influence in the Middle East on Thursday, a day after Russian President Vladimir Putin said of longstanding U.S.-Saudi ties, “everything changes.”

The Kremlin put on a lavish official welcome for Saudi Arabia’s King Salman. Mr. Putin greeted him in the Kremlin’s ornate Andreyevsky reception hall, following an arrival broadcast on Russian national television.

...