Carter was the first notable passing quarterback to play at Brigham Young University, whose football program became well-known for producing great passers. While at BYU, Carter set six national, 19 conference, and 24 school records and was an academic All-American. Carter began his college career under first-year coach Tommy Hudspeth, who was taking over a program that had produced two winning seasons in the previou ten years. BYU went 3–6–1 that first year, but Carter threw for over a thousand yards. The Cougars won the Western Athletic Conference championship in 1965, going 4–1 in WAC play and 6–4 overall to win the first conference championship in program history. The following year, the Cougars won eight out of ten games despite finishing second in the WAC, and Carter threw for over two thousand yards.  Notably, the success BYU experienced with Carter at quarterback influenced then-assistant coach LaVell Edwards to adopt a pass-oriented offense after replacing Hudspeth as head coach in 1972.
Carter was drafted by the Chicago Bears in the 6th Round of the 1967 draft  and was traded to the Cincinnati Bengals after the 1969 season. He led the NFL in pass completion percentage in 1971 and was third in overall passing. His best game of that season was the opener, in which the Bengals defeated the Philadelphia Eagles 37-14. Carter completed 22 of 30 attempts for 273 yards, three touchdowns and no interceptions. The following year, he split time with Ken Anderson before Anderson took sole possession of the starting job. In 1973 the Bengals decided to go with Anderson as the starting quarterback, but Carter had to miss the entire season due to a broken collarbone. In 1974 Carter left the Bengals for the Chicago Fire of the World Football League.
Carter was the WFL's leading passer in 1974 until an injury sidelined him in week eleven. He finished the season with 358 attempts completing 195 for 2629 yards. He threw 27 touchdown passes and was intercepted 16 times. The Fire offense in 1974 is compared today to the West Coast Offense.
In 1975, he was signed by the San Diego Chargers, then traded to the Bears during the season. He retired from the Bears after the 1976 season.
Carter was a highly intelligent quarterback, who blossomed in Cincinnati under the west coast system implemented by Bill
Walsh, then the Bengals' quarterbacks coach and later head coach of the San Francisco 49ers. In his first stint with the Bears Carter earned a master's degree from Northwestern, and while in Cincinnati with the Bengals taught statistics and mathematics at Xavier University.