Saeid Mehri scored the solitary goal of the match in the first extra time at the Takhti Stadium in Jam.
Elsewhere, Havadar defeated Azadegan League side Esteghlal Mollasani 2-1 in its away match.
Mohammad Chaharmahali gave the visiting team the lead in the 39th minute but Mohammad Amin Saeidavi leveled the score in the 66th minute.
Havadar captain Masoud Shojaei scored the winner in the injury time.
Persepolis will play Gol Gohar Sirjan in Tehran on Sunday, while titleholder Nassaji face Paykan in Ghaemshahr.
Hazfi Cup is the Iranian football knockout cup competition, run by the Iranian Football Federation.
Esteghlal is the most successful club with seven titles.