The Reds hosted the match in an empty Azadi Stadium as they were banned from having spectators by Iran's Premier League (IPL) Disciplinary Committee.
The breakthrough came after 83 minutes through Persepolis' Sepehr Heidari, whose mistake let Esmail Sharifat equalize the match only two minutes later.
The disappointing draw was Persepolis' second in as many matches. The Reds were held to a 3-3 draw at Mes Kerman in their season opener last Thursday.
Earlier on Friday, Steel Azin defeated Saba 4-3 to claim the IPL top spot with six points out of two matches.
A strike by Ali Karimi and a hat trick by Gholami, which was completed in injury time, sealed Steel Azin's second straight win, which followed last week's victory over Pas.
In other Friday matches, Saipa beat newly promoted Shahin 1-0 to celebrate its first win after last week's shocking 3-0 defeat to Zob Ahan. Malavan defeated Rah Ahan 2-1 while Abumoslem drew 0-0 with Pas.