On January 21, 2015, Norris was named by MLB as the third-best LHP prospect in baseball. In 2015, MLB named Norris as the number 1 Blue Jays' prospect, the number 3 left-handed pitching prospect in baseball, and the 17th best prospect overall. He opened the 2015 season in the starting rotation for the Blue Jays. After his fifth start, Norris was optioned to Triple-A Buffalo on May 1.
On July 30, 2015, Norris was traded to the Detroit Tigers along with Matt Boyd and Jairo Labourt in exchange for David Price. In his debut for the Tigers on August 2, Norris pitched 7⁄3 innings and struck out five Baltimore Orioles hitters, allowing one walk, four hits, and just one earned run on a solo home run by Chris Davis in the fourth inning. Following the home run to Davis, Norris retired the next 12 batters in a row, earning the win in a 6–1 Tigers victory.
On August 19, 2015, Norris became the first Tigers pitcher to hit a home run in his first career plate appearance, the first pitcher to do so since Tommy Milone in 2011, and the first American League pitcher to do so since Esteban Yan in 2000. He also became the first Tigers pitcher to homer since Jason Johnson did so on June 8, 2005, and the first American League pitcher to ever hit a regular-season home run at Wrigley Field. Norris exited the game in the fifth inning with a right oblique strain, and was placed on the disabled list the following day.
On October 19, 2015, Norris announced through his Instagram account that he had been diagnosed with thyroid cancer. He was diagnosed the previous April, and with a doctor's consent, he continued playing and delayed treatment until the end of the season. On October 29, Norris announced he was cancer free following a successful surgery to remove a malignant growth from his thyroid.