Karrar Jassim put the host into the lead in the 57th minute from the penalty spot.
Mehdi Torabi equalized the match in the 84th minute and the two teams went to extra time 1-1.
Torabi was on target once again in the 97th minute and Arash Rezavand made it 3-1 in the 107th minute.
Saipa joined Sepahan and Damash Gilan who have already qualified for the Hazfi Cup’s semis.
The Hazfi Cup is the Iranian football knockout cup competition, run by the Iranian Football Federation.
Esteghlal is the title-holder and most decorated team with seven titles.