The Royals had grabbed a 4-3 lead in the top of the 12th on Billy Butler's RBI double off Jordan Norberto.
But Broxton (0-1) and the Royals' defense blew the advantage in the next half-inning.
With one out, pinch-hitter Seth Smith reached base when shortstop Alcides Escobar booted his grounder. Broxton then loaded the bases with walks to Jemile Weeks and Eric Sogard, and Coco Crisp drove in the equalizer with a single.
Oakland pushed across the winning run when Broxton followed by hitting Yoenis Cespedes and Johnny Gomes on consecutive pitches.
Gomes also socked a two-run homer for the Athletics.
Andrew Carignan (1-1) earned the victory after getting the final out in the top of the 12th as Oakland took the rubber match of its three-game series against the Royals.
Butler and Alex Gordon each delivered solo homers and two RBI for Kansas City, which wrapped up a 3-3 season-opening road trip.
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