Arash Borhani opened the scoring for Esteqlal after 16 minutes, seven minutes before Reza Enayati doubled the lead.
Esteqlal Ahvaz returned to the match only four minutes after the break when Reza Ma'quli made the score 2-1.
Amir Khalifeh-Asl equaled the match in the 86th minute and Omid Ravankhah silenced 15,000 Esteqlal fans at Tehran's Azadi Stadium with a last-minute goal.
Despite the home defeat, Esteqlal remained on top of Iran's Premier League with 22 points, one point clear of Peykan, which came back from two goals to draw 2-2 with Pas.
Also on Monday, Ali Karimi scored twice for Steel Azin in a 2-2 draw with Saipa while Malavan beat bottom-placed Abumoslem 3-1.