The incident took place 10 minutes into the Championship tie.
Members of the emergency services rushed to the spectator's aid. Paramedics were unable to revive him and the game was abandoned.
Greenock Morton said: "All at Greenock Morton Football Club would like to take this opportunity to extend our heartfelt sympathy to all family and friends of the gentleman concerned."
A spokesperson for Queen of the South FC said: "We would like to echo those comments and also pay tribute to the efforts of those involved in the attempts to save the man's life, including our own club doctor Guy Beaumont who was at the match and assisted Morton's Doctors.
A report will be submitted to the procurator fiscal.