As one of 11 children of an Alliance, Ohio pipefitter, Dawson quickly showed his leadership ability, serving as team captain for Alliance (OH) High School's football, baseball and basketball teams. During his years on the school's football team, he twice earned All-State recognition: as a linebacker in 1951, followed the next year at quarterback. In his latter role, he threw for a school-record 1,615 yards.

During the recruiting process, Dawson had to choose between Ohio State University and Purdue. While he was reluctant to take over Woody Hayes' split-T offense with the Buckeyes, the true reason for his selection of Purdue stemmed from the chemistry he had established with a Boilermaker assistant coach, Hank Stram, beginning a friendship that would last for more than a half century. During three seasons with the Boilermakers, Dawson threw for over 3,000 yards, leading the Big Ten Conference in that category during each campaign.

An equally fulfilling relationship for Dawson had begun during his junior year at Alliance High School, when he started dating Jacqueline Puzder, a recent transplant from Cleveland, Ohio who was one grade behind him. Over the next two years, the two became closer, and after Puzder went to visit Dawson during his freshman year at Purdue, the two were secretly married. Two weeks prior to her graduation, she gave birth to the couple's first child.

As a sophomore in 1954, Dawson put together an outstanding first season as the NCAA's leader in pass efficiency, while also playing defense and serving as the Boilermaker kicker. Blessed with a strong offensive line, he threw four touchdown passes in a 31-0 victory over the University of Missouri, then later engineered a huge upset of the University of Notre Dame. The Fighting Irish had entered the contest in the midst of a 13-game winning streak.

Despite his status as a first round draft pick, Dawson was unable to make an impact with the Pittsburgh Steelers. Following his rookie campaign, his status in the Steel City became even more tenuous when the Steelers acquired future Hall of Famer Bobby Layne. Failing to dislodge the colorful signal caller, Dawson was then traded to the Cleveland Browns on December 31, 1959.

After encountering similar problems in battling Browns' quarterback Milt Plum, Dawson was released, having completed only 21 passes for 204 yards and two touchdowns in his five seasons of NFL play. However, he soon found his calling when he signed with the American Football League's Dallas Texans on June 30, 1962. The move reunited him with Stram, who was beginning his third year as the Texans' head coach.

In that first season, Dawson led the league in touchdowns and yards per attempt, and was The Sporting News' selection as the AFL MVP. He also led them that year to the first of three league titles in a thrilling double-overtime victory over the two-time defending champion Houston Oilers. Dawson ran a ball-control offense in the 17-14 win, and tossed a 28-yard touchdown pass to halfback Abner Haynes.

A pinpoint passer, Dawson's mobility helped him flourish in Stram's "movable pocket" offense. He would win four AFL passing titles and was selected as a league All-Star six times, ending the 10-year run of the league as its highest-rated career passer.

While he threw for more than 2,000 in each of the previous seven campaigns, Dawson's 1969 season would be his most memorable by making a dramatic comeback from a knee injury suffered in the season's second game. The injury was first feared as season-ending, but after missing five games, Dawson went on to lead the Chiefs to road playoff victories over both the defending champion New York Jets and the Oakland Raiders.

He then capped his year with MVP accolades in Super Bowl IV, the last game ever played by an American Football League team. In the game, Dawson paced the Chiefs to a win over the NFL's heavily-favored Minnesota Vikings by completing 12 of 17 passes for 146 yards. The performance was especially notable given the fact that he had been linked to a gambling investigation in the days leading up to the game.

With the league's absorption into the National Football League in 1970, Dawson earned one final honor from the league as a member of the second team All-time All-AFL Team. He is also a member of the Chiefs' Hall of Fame and the Pro Football Hall of Fame. He would earn Pro Bowl honors following the 1971 NFL season, then ended his career in 1975, having thrown for 239 touchdowns and over 28,000 yards.

In 1987, Dawson was recognized for his play with the Chiefs with his induction into the Pro Football Hall of Fame.

Dawson currently works on the Chiefs radio broadcast team, in addition to his duties as sports director for KMBC-TV in Kansas City. From 1977 to 2001, he served as the host of HBO's Inside the NFL.