Bay belted a two-run homer in the top of the first inning, while Mintkiewicz hit an RBI double in the third and his RBI single in the seventh snapped a 6-6 tie in the seventh.
Four others drove in one run and three others had two hits for the Pirates, who swept the three-game series at Minute Maid Park following a five-game losing streak.
The winner after two scoreless relief innings was Franquelis Osoria (4-3).
Damaso Marte hurled a scoreless ninth to earn his fifth save.
Houston's Mark Loretta had three hits and two RBI.
Five others drove in one run.
Michael Bourn had two hits and Kaz Matsui scored three times for the Astros, whose four-game win streak ended.
The losing pitcher was Geoff Geary (2-2).
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