Majka, who was third overall in the 2015 edition of this race, won on a solo breakaway up the final special-category ascent. The BORA-hansgrohe rider finished the 175-kilometer (108.7-mile) stage from Ecija to Sierra de La Pandera in 4 hours, 42 minutes, 10 seconds.
Miguel Angel Lopez was 27 seconds back in second.
Vincenzo Nibali beat Froome to a third-placed finish for the stage, grabbing a four-second bonus that shaved Froome’s lead over the Italian to 55 seconds.
Nibali attacked Froome on the climb, but the four-time Tour de France winner kept him close and then closed the gap with 2 kilometers (1.2 miles) left on the punishing final ascent.
Froome is trying to win the Vuelta for the first time after finishing the grand tour runner-up on three occasions.
Majka said he relished the win after dropping out of overall contention in earlier stages when he felt ill.
“So I am really happy, not for me, but for my teammates because they did a really great job,” Majka said. “I was really confident. I knew I had good legs.”
Wilco Kelderman moved into third place overall at 2:17 behind Froome. Esteban Chaves fell to fifth behind Ilnur Zakarin.
Riders face another tough test in the southern mountains on Sunday. The 15th stage is a 129.4-kilometer (80.4-mile) ride over two category-one climbs before a special-category finish at Alta Hoya de la Mora.
The three-week race ends in Madrid on Sept. 10.
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