Bixler finished 3-for-5 with a solo shot while Maxwell provided a two-run blast for Houston, which built a 9-2 lead but then had to weather a five-run Royals uprising in the ninth.
Brian Bogusevic went deep as Houston took the opener of a three-game series.
J.A. Happ (5-7) earned the victory, yielding a pair of runs on four hits. The left-hander walked one and struck out three in six innings.
Reliever Brett Myers almost blew the big lead, surrendering five runs on eight hits in the ninth.
Xavier Cedeno earned his first major league save by inducing Mike Moustakas to foul out with the bases loaded to end the game.
Yuniesky Betancourt drove in two runs during the late rally and Billy Butler knocked in a pair for Kansas City.