The 1970 Vikings posted the NFL's best regular season record at 12-2, but lost in an NFC Divisional playoff game to the San Francisco 49ers at home. In 1971, Cuozzo could not hold on to his starting job, sharing duties with Norm Snead and Bob Lee. The Vikings went 11-3 in the regular season and lost in the divisional playoffs at home to the eventual Super Bowl champion Dallas Cowboys.
After the 1971 season, the Vikings reacquired Fran Tarkenton from the New York Giants, with Snead and the Vikings' leading receiver from 1971, Bob Grim, going to the Big Apple. Tarkenton played his first six seasons (1961-66) in Minnesota, coinciding with the Vikings' first six seasons in the NFL.
Cuozzo was traded to the St. Louis Cardinals in the deal which sent fleet wide receiver John Gilliam to the Vikings. Cardinals coach Bob Hollway was familiar with Cuozzo, having served as Minnesota's defensive coordinator under Bud Grant prior to leaving for St. Louis in 1971.
Cuozzo was part of a chaotic four-quarterback rotation with Jim Hart, Pete Beathard, and Tim Van Galder in 1972, but when Don Coryell took over as Cardinals coach in 1973, he named Hart the undisputed starter, and he would hold the job into the 1980s.
In 1990, his oldest son Gary Jr., a/k/a Chip, was murdered in Miami during a drug deal, and Cuozzo gave talks to teens about avoiding drugs. He served as national chairman of the Fellowship of Christian Athletes from 1995 to 1998.