Saad al Jabri was America’s closest antiterrorism contact in the Arab world for more than 15 years, beginning after the 9/11 attacks. The Saudi spymaster had informants, access to billions of government dollars, close relationships with key members of the royal family and an ability to get things done that Americans found rare among their connections in the Middle East.

Now Mr. Jabri is an international fugitive. He is on the run from Saudi Arabia, where officials allege that a group of men he led while working for the Interior...