Minshew attended Brandon High School (Brandon, Mississippi), in Brandon, Mississippi where he played quarterback for the Brandon Bulldogs from 2011–2014. He played sparingly during his freshman year, but he became the starting quarterback his sophomore year, a job he kept through his senior year. During his sophomore year in 2012, he led the Bulldogs to the 6A Mississippi Championship game, where the Bulldogs lost, 23–31. Minshew led the Bulldogs to a South 6A championship his senior year in 2014. He was rated as a three-star prospect and the 70th-best pro-style quarterback in his class coming out of high school by the 247Sports.com Composite, which aggregates the ratings of the major recruiting services. Minshew committed to play football at Troy in December 2014. He graduated from high school early and attended school at Troy on an academic scholarship for a semester before deciding to transfer to Northwest Mississippi Community College.
Minshew transferred to East Carolina University in 2016 where he played in seven games while starting two at QB for the Pirates. In 2017, he split time with Duke University transfer Thomas Sirk as East Carolina's quarterback. Minshew began to show his talents late in the season by passing for over 350 yards in 3 consecutive games (463 yards and 3 touchdowns against Houston, 444 yards and 4 touchdowns against Cincinnati and 351 yards against Memphis). Minshew finished the year with 2140 yards passing and 16 touchdowns for the Pirates. Minshew received his bachelor's degree from East Carolina University in December 2017.
Minshew enrolled as a graduate student at Washington State in the fall of 2018. He became immediately eligible under the NCAA's graduate transfer rules.
Minshew helped lead Washington State to a school-record-breaking 11 wins in 2018. In his third game as the school's starting quarterback, he passed for a career high and school record 470 yards and a 78.9% completion rate in a 59–24 victory over Eastern Washington. On November 17, he again broke school records with 473 passing yards and seven passing touchdowns against Arizona.
At the end of the 2018 regular season, Minshew led FBS in pass completions (433), pass attempts (613), passing yards per game (367.6) was second in passing yards (4,477) and finished in the top five with 38 touchdowns. During Washington State's Alamo Bowl win over Iowa State, Minshew claimed the Pac-12 single season record for most passing yards, a record previously held by Jared Goff.
Minshew was awarded the Johnny Unitas Golden Arm Award as the nation's top senior- or fourth-year quarterback. He was named the Pac-12 Offensive Player of the Year for 2018, joining Rueben Mayes (1984 and 1985), Steve Broussard (1989), Drew Bledsoe (1992), Ryan Leaf (1997), and Jason Gesser (2002) as the only Washington State offensive players to win the award since the it began in 1975. Minshew finished fifth in voting for the Heisman Trophy, becoming the ninth player in Washington State history to place in the top 10 for the award, and the highest WSU finisher since Ryan Leaf, who was third in Heisman voting in 1997.