Murray was born in Bedford, Texas, to Kevin Murray and Misun Henderson. He attended Allen High School in Allen, Texas. As a senior in 2014, he was the Gatorade Football Player of the Year. His team won three straight state championships and 43 games in a row. He missed one start during this streak and finished his career with a perfect 42–0 record as a starting quarterback.

Murray was ranked by ESPN, Scout.com, and 247Sports as a five-star recruit and the best dual-threat quarterback in his class. In May 2014, he committed to Texas A&M University to play college football and college baseball. He officially signed with them on February 4, 2015.

As a true freshman in 2015, Murray competed with Kyle Allen before the season for the starting quarterback position. Allen won the job, with Murray appearing in games as a backup (primarily on plays involving the wildcat formation). After the team lost two straight games in October, the competition was opened again prior to the South Carolina game, and Murray won the starting quarterback position. In his first start, Murray threw for 223 yards and rushed for 156 yards with a passing touchdown and a rushing touchdown, making him and Cam Newton the only Southeastern Conference quarterbacks in the past 20 years to gain over 100 yards passing and rushing, including a passing touchdown and a rushing touchdown in their first career start. On December 24, 2015, Murray announced that he was transferring to the University of Oklahoma. Per NCAA transfer rules, he had to miss the 2016 season.

Murray opted out of the MLB draft after deciding to attend Texas A&M. He was initially set to play as an infielder for the Texas A&M Aggies baseball team in the 2016 season until he announced his departure from Texas A&M on December 17, 2015.

In the 2017 season, Murray was a backup quarterback to Baker Mayfield. He got the opportunity to play in some games in relief of Mayfield. He made his season debut in the season opener against UTEP. In the 56–7 victory, he came into the game in the second half and finished 10-of-11 for 149 yards and a touchdown. Two weeks later, against Tulane, he hit all three pass attempts for 103 yards and a touchdown. On November 25, against West Virginia, he started the game after Mayfield had to sit out some of the game due to disciplinary actions in the previous game against the Kansas Jayhawks. In the game, he hit both pass attempts for 52 total yards and a touchdown while contributing 80 rushing yards on three carries.

Murray played for the Oklahoma Sooners baseball team in 2017 as a left fielder. He had a .122 batting average and six runs batted in (RBIs) in 27 games. After the 2017 season, he played collegiate summer baseball for the Harwich Mariners of the Cape Cod Baseball League. In 2018, playing as a centerfielder, Murray batted .296 with 10 home runs, 47 RBIs, and 10 stolen bases.

Without Mayfield, Oklahoma considered Murray and Austin Kendall to start at quarterback in 2018. Murray was chosen as the starting quarterback. He threw for more than 4,000 yards and 40 touchdowns in the 2018 regular season. In December 2018, Murray was awarded the Heisman Trophy as the best collegiate football player in the country, giving Oklahoma back-to-back Heisman winners. Murray helped lead Oklahoma to a Big 12 Championship and an appearance in the College Football Playoff. In the 2018 Orange Bowl against Alabama, he had 308 passing yards and two passing touchdowns as the Sooners fell by a score of 45–34. Murray became only the third Heisman Trophy winner to not be awarded consensus All-American honors.

Despite Murray’s reported intention to continue his football career, the Oakland Athletics selected Murray with the ninth overall selection of the 2018 MLB draft. He signed a contract with the Athletics which included a $4.66 million signing bonus. Prior to his junior year with the football team, Murray said that he would report to spring training in 2019 to begin his professional baseball career. However, following his Heisman Trophy-winning 2018 season, he would forgo his senior year and enter the NFL Draft. The A’s retain his rights, should he return to baseball.

In January 2019, Murray announced that he would enter the 2019 NFL Draft. On February 11, he announced his decision to focus on his football career, forgoing baseball.

Murray was drafted first overall by the Arizona Cardinals in the 2019 NFL Draft, becoming the shortest quarterback drafted in the first round (Johnny Manziel was 5’11” 3/4″). When leading Texas Tech’s football program, Cardinals head coach Kliff Kingsbury said that he would hypothetically spend the draft’s top selection on Murray. The pick made Murray the first player to be drafted in the first round of both the NFL and MLB drafts, the first time that two quarterbacks from the same university have gone in the first round in successive seasons after Baker Mayfield was selected first in the 2018 NFL Draft and the first quarterback under 6 feet selected number one in an NFL draft.

Murray played his first NFL game on September 8, 2019 against the Detroit Lions, finishing with 308 passing yards, two touchdowns, and an interception as the game ended in a 27–all tie. Murray led the Cardinals to a comeback in the fourth quarter from a 16-point deficit, joining Jake Plummer as the only Cardinals rookie quarterbacks to throw for 300 yards and two touchdowns since the AFL-NFL merger in 1970. During Week 2 against the Baltimore Ravens, Murray finished with 349 passing yards as the Cardinals lost 17–23. In Week 4 against the Seattle Seahawks, Murray threw for 241 yards and a pick-six and rushed four times for 27 yards and his first NFL rushing touchdown in the 27–10 loss. In the next game against the Cincinnati Bengals, Murray threw for 253 yards and rushed for 93 yards and a touchdown in the 26–23 victory. During Week 6 against the Atlanta Falcons, Murray threw for 340 yards and three touchdowns and rushed 11 times for 32 yards in a narrow 34–33 victory. He was named the NFC Offensive Player of the Week for his performance. In Week 10, against the Tampa Bay Buccaneers, he had 324 passing yards, three touchdowns, and one interception in the 30–27 loss. In Week 11 against the San Francisco 49ers, Murray threw for 150 yards and two touchdowns and rushed for 67 yards and a touchdown in the 36–26 loss. In Week 16 against the Seattle Seahawks, Murray threw for 118 yards and a touchdown and rushed for 40 yards before exiting the game due to a hamstring injury. Without Murray, the Cardinals won the game 27–13. Before the regular season finale against the Los Angeles Rams, Murray was listed as a game time decision with a hamstring injury and ultimately played in the game despite being hurt. During the game, Murray threw for 325 yards, two touchdowns, and two interceptions during the 31–24 loss. Afterwards, Cardinals’ head coach Kliff Kingsbury said that “It’s the most proud I’ve been of him all season.”

In Week 1 against the San Francisco 49ers, Murray finished with 230 passing yards, one passing touchdown, an interception to go along with 91 rushing yards and a rushing touchdown in the 24–20 win. In Week 2 against the Washington Football Team, Murray passed for 286 yards, one passing touchdown, and one interception to go along with eight carries for 67 rushing yards and two rushing touchdowns in the 30–15 victory. During Week 5 against the New York Jets, Murray finished with 380 passing yards, 31 rushing yards, two total touchdowns, and one interception as the Cardinals won 30–10. On October 14, 2020, Murray was named the NFC Offensive Player of the Week for his performance in Week 5. In Week 7 against the Seattle Seahawks on Sunday Night Football, Murray completed 34 of 48 passes for 360 yards, three touchdowns, and an interception plus ran 14 times for 67 yards and scored a rushing touchdown as the Cardinals beat the previously undefeated Seahawks 37–34 in overtime. Murray was also named NFC Offensive Player of the Week for his performance in Week 7. In Week 9, against the Miami Dolphins, he had 283 passing yards, three passing touchdowns, 106 rushing yards, and one rushing touchdown in the 34–31 loss.