The two teams failed to score any goals in the 90 minutes of regulation and the ensuing 30 minutes of overtime in front of 65,000 spectators at Tehran's gigantic 100,000-seater Azadi Stadium on Friday night.
Saba, coached by Rasoul Korbekandi, was playing with 10 men from the 91st minute after defender Morteza Kashi was sent off for a second caution.
"Playing against Saba FC was like a real final game. We defeated a great team," said Persepolis manager Ali Daei.
"Unfortunately, we failed to take advantage of several opportunities we created. We could have scored three goals in the last 15 minutes of the game," he added. "Persepolis deserved to win the game."
This is the fifth time the Reds have reached the Iranian cup elimination final.
Daei's men must now meet Gostaresh Foulad, which thrashed Zob Ahan at the Yadegar-e Imam Stadium in Tabriz earlier on Friday.
The cup elimination final is a knockout cup competition in Iranian football, run by the Football Federation of the Islamic Republic of Iran (FFIRI).