Mike Moustakas was retired on a pop-out and George Kottaras forced a runner at home on a grounder, but Maxwell then bashed a pitch 430 feet to left field for his mammoth, game-winning grand slam.
Tim Collins (3-6) got the win for the Royals, who took the rubber match of a three-game series and pulled to within 3 1/2 games of a American League wild-card spot with Texas ahead of them.
The Rangers, meanwhile, fell 1 1/2 games behind Cleveland for the second AL wild-card spot.
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