Cocozza began his career with Celtic but failed to make a senior appearance for the club, leaving in November 2001 to sign for Dundee United. Cocozza made his debut in a televised match against Celtic but made only one further substitute appearance and failed to earn an extension to his short-term deal. After being released in April 2002, Cocozza moved to Forfar Athletic at the start of the 2002-03 season, although had to wait until February 2003 to make his debut. After playing in three matches, Cocozza was released by the end of the season and he left the professional game, signing for Johnstone Burgh. The following year, Cocozza moved to Maryhill, where he stayed until summer 2005.