The 22-year-old Walker, looking to lock up a rotation spot, struck out six and walked one in six more shutout innings. He gave up two hits.
Zunino launched his team-leading fourth home run and Robinson Cano had two hits for Seattle, including an RBI double.
Chicago left-hander Travis Wood, also competing for a starting slot, allowed three runs and five hits in five innings. He struck out three and walked two.
Walker, who has made 11 major league starts, put two runners on in the sixth but escaped the jam by getting Dexter Fowler to bounce into a double play.
