 He began his NFL career on the Baltimore Colts as a backup to Johnny Unitas.When Unitas was injured in 1965, Cuozzo replaced him until getting sidelined by injury as well, forcing coach Don Shula to use running back Tom Matte as an emergency quarterback.On March 6, 1967, the Colts traded Cuozzo to the expansion team New Orleans Saints, as part of a deal that also sent offensive lineman Butch Allison to the Saints in exchange for a 1967 first round pick (#1-Bubba Smith), a 1967 third round pick (#54-Norman Davis), a 1969 seventh round pick (#163-Gary Fleming) and center Bill Curry.