Fattahi lost to Meerim Momunova from Kyrgyzstan in the final match of the 60kg.
Yazdani also won a bronze medal at the 55kg weight class.
Iran has participated in the competition with six female wrestlers.
Belt Wrestling is a form of wrestling that is one of the oldest historically recorded sports. It involves contestants aiming to knock each other over by grappling with a belt.
The competitions is underway in Nur-Sultan, Kazakhstan from December 20 to 23.