As a sophomore in 1962, he could only play in five games because of a knee injury, but still managed 905 passing yards, a 54% completion rate and nine touchdowns. As a junior in 1963 he already owned most of Cal's all-time quarterback records.
In his three seasons as a starter he would never play on a winning team. His senior season in 1964 saw him complete 185 of 308 passes for 2,121 yards and 13 touchdowns, and even with a losing 3–7 record, he was recognized for his talent and contributions by being named first team All-American over other winning quarterbacks. He would also receive the W. J. Voit Memorial Trophy, given to the best player on the Pacific Coast and the Pop Warner Trophy, given to the best senior player.
He finished his career with 4,501 passing yards (a Pac-8 record), and most of Cal's all-time passing records, including:
Morton was drafted by Dallas Cowboys in the first round of the 1965 NFL Draft (fifth overall). He spent his first 4 seasons as the backup for Don Meredith, but still received opportunities to play due to different injuries suffered by Meredith. In 1969 he became the starter at quarterback after Meredith's surprising retirement, but suffered a separated right shoulder in the fourth game of the season, that would require surgery at the end of the year and that affected his passing percentage (dropping from 71.1% to 53.6%).
In 1970, although he was bothered most of the season recuperating from offseason right shoulder and right elbow surgeries, he led the Cowboys to Super Bowl V where the team lost a close game to the Baltimore Colts. This result created one of the most famous quarterback controversies in NFL history, when in 1971Tom Landry started alternating Morton with Roger Staubach, reaching an extreme point against the Chicago Bears, where they alternated on each play. After this famous game, Landry settled on Staubach and the Cowboys went on a 10-game winning streak that included Super Bowl VI.
Morton played all of the 1972 regular season because of a separated shoulder suffered by Staubach, but he was replaced by him during a playoff game against the San Francisco 49ers. Staubach entered a hopeless situation and would go on to throw two touchdown passes in the last 90 seconds to win the game 30-28, eventually sealing Morton's fate with the team.
At age 34 he revived his career with the Denver Broncos, finishing the season as the second rated passer in the AFC. Although he was suffering in the playoffs from a swollen left hip that needed to be drained, he was able to overcome the injury and become the first quarterback in NFL history to start the Super Bowl for two different teams (Dallas in Super Bowl V and Denver in Super Bowl XII), a feat that has only been equaled by Kurt Warner and Peyton Manning. Morton, however, holds the unique distinction of being the only quarterback to have started two different team's inaugural Super Bowl appearances, for the Cowboys and Broncos, while Warner's Rams and Manning's Colts and Broncos had several past Super Bowl appearances in their franchise history. After the 1977 season, Morton was named the NFL Comeback Player of the Year and selected All-AFC by the Sporting News.
Morton's best statistical season came in his 17th season (1981), when he threw for 3,195 yards and 21 touchdowns, compiling an excellent 90.5 passer rating and setting several records. He was a downfield passer not known for his mobility, and is one of the all time leaders in yards per completion. He also briefly held the record for the most passes completed in a row.
He wore number 7 for the Broncos, and retired just before the arrival of John Elway in 1983, who also wore the same number and has had it since retired in his honor. He still remains as the third all-time passing yards leader in team history with 11,895 and his regular-season record was 50 wins and 28 losses in five seasons.
Although Roger Staubach replaced him for the Dallas Cowboys in 1973 and would go on to great success, Morton played seven more NFL seasons than Staubach. The Cowboy teammates graduated from college the same year, but Staubach began his NFL career four years later following his active service in the U.S. Navy, and chose to retire after the 1979 season, while Morton played for three more seasons until 1982.