Norris attended Balboa High School in San Francisco and was drafted by the Oakland Athletics in the first round (24th overall) In January 1973.

Norris spun a complete-game shutout in the first round of the playoffs against the Kansas City Royals and pitched well in the AL Championship Series against the New York Yankees despite getting hit with a loss there.

A number of baseball historians and statisticians have blamed this on manager Billy Martin overworking him and the other members of the 1981 staff.

In 2006, baseball writer Rob Neyer estimated that Norris threw 131 pitches per complete game in 1981–a heavy workload for a young pitcher even then.

In 2007, he appeared with Dave Stewart, Mudcat Grant, and Vida Blue in a pre-game ceremony before a regular season game between the Texas Rangers and the Oakland Athletics.