He played college football for the California Golden Bears, receiving All-American honors, and was selected by the Cowboys fifth overall in the 1965 NFL draft.

He completed 185 of 308 passes for 2,121 yards and 13 touchdowns in his senior season in 1964, but 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.

In the balloting for the Heisman Trophy won by John Huarte of Notre Dame, Morton was seventh, ahead of Joe Namath of Alabama and Gale Sayers of Kansas.

After this famous game, Landry settled on Staubach and the Cowboys went on a 10-game winning streak that included a 24–3 victory in Super Bowl VI over the Miami Dolphins.

In 1972, Staubach suffered a separated right shoulder in the third preseason game against the Los Angeles Rams and Morton was named the starter.

For the first time since 1969, his arm had regained its strength, helping him register 185 completions (club record) out of 339 attempts (54.6%), 2,396 yards (fifth in the league), 15 touchdowns and 21 interceptions.

Staubach entered an apparently hopeless situation trailing 28–13 and threw two touchdown passes in the last 90 seconds to win the game 30–28,[8] eventually sealing Morton's fate with the team.

On April 10, after repeatedly asking for a trade, Morton signed a contract with the Houston Texans of the World Football League for their 1975 season.

[10] After acquiring Morton, the Giants traded their starting quarterback Norm Snead to the San Francisco 49ers in exchange for a third-round draft choice in 1975 and a fourth in 1976.

Although he suffered in the playoffs from a swollen left hip that needed to be drained,[14] he overcame the injury to become the first NFL quarterback to start the Super Bowl for two different teams (Dallas in V and Denver in XII), a feat only equaled by Kurt Warner, Peyton Manning, and Tom Brady.

He threw a franchise playoff record four interceptions in Denver's defeat against his former team, the Cowboys, which led to him being pulled in favor of Norris Weese.

Morton remains the third all-time passing yards leader in franchise history with 11,895 and his regular-season record was 50 wins and 28 losses in five seasons.

Following his playing career, Morton served as head coach for the Denver Gold of the United States Football League (USFL).