Norris (7-13) scattered four hits and walked a pair over six scoreless frames, striking out four.
Xavier Cedeno, Chuckie Fick and Wesley Wright teamed up for two shutout frames and Wilton Lopez collected his 10th save for Houston, which clinched its season-ending three-game set.
Jason Castro homered and Tyler Greene went 2-for-3 with a run-scoring single for the victors.
Chris Volstad (3-12) allowed only one earned run in absorbing the loss, surrendering five hits over seven full frames for the Cubs.
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