At Cal Poly, Norris was a roommate as well as a teammate of fellow major league pitcher Casey Fien.

[2] Norris was selected by the Houston Astros in the sixth round (189th overall) of the 2006 Major League Baseball draft.

[8] He made his major league debut on July 29, pitching three innings of relief against the Chicago Cubs.

He was the last pitcher to bat at Minute Maid Park prior to the Astros' transfer to the American League for the 2013 season.

Just before the deadline for clubs and players to exchange numbers for arbitration on January 18, 2013, Norris agreed to $3 million for the 2013 season.

[11] On July 31, 2013, Norris was traded to the Baltimore Orioles for L. J. Hoes and minor league pitcher Josh Hader.

[12] On the day after the trade, the Astros were already in Baltimore playing the Orioles, so Norris switched dugouts and won the game.

Norris had also had run-ins earlier in the season with Boston Red Sox catcher David Ross and Pittsburgh Pirates infielder Neil Walker, but the Hunter incident resulted in his first major-league ejection.

[20] He began the 2016 season in the Braves rotation, but had been moved to the bullpen by the end of April, as he had compiled a 1–4 record and 8.74 ERA in five starts.

[28] For the 2016 season — combining Norris’ overall stats from the Braves and Dodgers — he tied (with Clay Buchholz) for the major league lead in grand slams allowed, with three.

[32] After a strong performance in spring training, Norris won a spot in the Angels bullpen as their long reliever.

On July 30, 2017 Norris surrendered a walk-off grand slam to Steve Pearce to cap a seven-run ninth inning that lifted the Toronto Blue Jays to an 11–10 victory over the Angels.

This made him the first pitcher to allow two walk-off grand slams in one season since Francisco Rodriguez did in 2009 (six days prior on July 24, Norris surrendered a grand slam to Edwin Encarnación with the score tied in the bottom of the eleventh that gave the Cleveland Indians and 11–7 win over the Angels).

[35] On March 6, 2019, Norris signed a minor league contract with the Toronto Blue Jays that included an invitation to spring training.

On July 10, 2021, Norris signed with the Winnipeg Goldeyes of the American Association of Professional Baseball.