WASHINGTON—Republican Kevin McCarthy won election as House speaker on the 15th ballot after days of grueling negotiations that forced leadership to make concessions to the party’s far-right members up to the last moment of sometimes-heated discussions on the House floor.

The victory marked a personal achievement for Mr. McCarthy, 57, who had failed to secure the speakership more than seven years ago, and clawed his way to a win early Saturday morning after four days of repeated rejections.