He saw limited playing time behind Eagles starting quarterback Norm Snead. In 1966, Concannon led the Eagles to several early December wins that allowed Philadelphia to finish the season with a 9–5 record and a tie for second place in the Eastern Conference of the NFL. He played with Philadelphia from 1964 through 1966 and then was traded to the Chicago Bears for tight endMike Ditka. He played for five years with the Bears. In 1967, Concannon rushed for a career-high 279 yards, but he was injured during the 1968 season and never put up significant rushing numbers again. His most productive season as a passer came in 1970, when he set career marks with 385 attempts, 194 completions, 2130 yards, and 16 TDs; as well as throwing 18 interceptions. His performance quickly dropped off after that, as he was limited to just three games due to an injury in 1971, his last season with the Bears. He was traded to the Dallas Cowboys, and played on the practice squad for two seasons. He was picked up by the Green Bay Packers in 1974 and ended his career as a backup in 1975 with the Detroit Lions.
In March 1981, Concannon was arrested on charges that he delivered a kilogram (2.2 lb) of cocaine to an undercover drug agent in Schaumburg, Illinois, a suburb northwest of Chicago. A year later in March 1982, he was acquitted. He successfully argued entrapment after admitting he was in desperate financial straits, but Cook County Criminal Court Judge Earl Strayhorn admonished Concannon:
"Seldom have I been confronted with a situation where a man has displayed such woeful lack of good judgment, particularly when that person has been in the limelight . . . has been supposedly a leader of men."
During his NFL career, Concannon became a partner in a restaurant. He also dabbled in acting, with a cameo appearance in the original theatrical film MASH (1970), and as himself in the critically acclaimed TV movie Brian's Song (1971), the story of Chicago Bears teammates Brian Piccolo and Gale Sayers.