Iran denied U.S. accusations that it was behind Saturday’s strikes on Saudi oil facilities, as Riyadh and Washington rushed to respond to attacks that sent futures soaring and threatened to spark a broader clash with Tehran.

U.S. officials said on Sunday there are strong indications the fiery blasts were the result of cruise-missile strikes launched from Iraq or Iran, an assessment that contradicts claims by the Iran-backed Houthi rebels in Yemen that their drones struck the two Saudi oil plants. U.S. Secretary of State Mike...