Houston pitching recorded its second shutout of the season, with Norris (3-1) striking out eight while allowing three hits over six innings. He did not walk a batter.
Brett Myers finished off the shutout with a perfect ninth inning, giving him nine saves on the year.
The only run of the game came in the second when Brian Bogusevic came in front third as Justin Maxwell was grounding into a double play.
The hard-luck loser was James McDonald (2-2), who also struck out eight with four hits over eight innings.
Pittsburgh loaded the bases with nobody out in the sixth, but Norris struck out two and got another out on a shallow fly ball to right to get out of the inning.