The first joint patrol by Russian and Chinese bombers aimed at testing the air defenses of U.S. allies in northeast Asia triggered a rare clash as South Korean fighters scrambled to ward off what Seoul said was an intrusion into its airspace.

South Korean pilots fired warning shots and flares Tuesday to repel a Russian command-and-control plane that was part of the bomber group, Seoul’s Defense Ministry said. Other Chinese and Russian warplanes passed several miles outside South Korean airspace, officials said.

The...