Amir Ghafour led Iran’s scorers with 14 points. Mohammadjavad Manavinezhad finished with 13.
Osniel Melgarejo with his personal high at the championship – 22 points– was top scorer for Cuba. Gutierrez added 15.
The Asian giant, who had defeated Puerto Rico and Bulgaria earlier, will face Poland on Monday.
Iran is in Pool D along with Bulgaria, Puerto Rico, Poland, Finland and Cuba.
The 2018 championship is the 19th staging of the FIVB Volleyball Men's World Championship, contested by the senior men's national teams of the members of the Fédération Internationale de Volleyball (FIVB), the sport's global governing body.
The final tournament is being held in Italy and Bulgaria from September 9 to 30. For the first time the tournament is jointly-hosted by more than one country.
The final six will be hosted by Italy at the Pala Alpitour in Turin.