Norris (3-2) held the Mariners to one run and the Astros bullpen limited the Mariners to one run in the final four innings.
Jose Veras pitched a perfect ninth for the Astros in earning the save.
Matt Dominguez had two hits for the Astros, who struck out 15 times as a team. Justin Maxwell drove in a run and Marwin Gonzalez homered.
Kendry Morales hit a home run for the Mariners. Hisashi Iwakuma (2-1) was handed the loss after an otherwise solid start. Iwakuma allowed three runs -- two earned -- on six hits and struck out 11 in five innings.