Iran-backed Houthi rebels in Yemen fired a barrage of missiles into Saudi Arabia on Wednesday, an early indication that Iran’s allies in the Middle East are likely to flex their muscles in a show of support for their patron—risking a wider conflict.

Saudi Arabia’s air-defense systems intercepted four ballistic missiles, two of which targeted Riyadh, and two launched towards the southern cities of Jizan and Najran, according to the Saudi Press Agency.

The...