Sen. Thad Cochran defeated a tea-party-backed challenger in a runoff Tuesday for the Mississippi GOP nomination, a remarkable electoral comeback that was a tribute to the power of incumbency even in today's anti-Washington political climate.

It was a big victory for the GOP establishment, which rallied to the side of the 76-year-old Mr. Cochran after he was forced into a three-week runoff campaign against state Sen. Chris McDaniel that focused on his ability to bring federal aid to the impoverished state and included a rare...