Barnett missed out entirely on Arsenal's 1970–71 Double-winning season but after an injury to Wilson late on in the 1971–72 season, played against Leeds United in the 1972 FA Cup Final. He was blameless for the goal which saw Arsenal lose 1–0 in a match better known for ill-discipline rather than the quality of play. Barnett started the 1972–73 season as Wilson struggled to recover from injury, playing 25 matches, but Wilson always assumed the goalkeeper's jersey ahead of Barnett when fully fit.