Penalties from Salem Khamis and Ali Al Hawasin and a Milad Midavoodi strike had put the hosts into a commanding 3-0 lead by the 66th minute.
But Saba responded as first skipper Fereidoun Fazli scored in the 68th minute and two goals in the space of a minute from Shahin Shafiei and Mohammad Nori levelled the scores.
Mohsen Yousefi then struck late on before Mohammad Hamrang scored to round off the stunning fightback and secure all three points and ensure Saba finish third in Group A.
The game was stopped for eight minutes soon after play got underway due to floodlight failure but once the match was restarted Saba had two quick chances.
First, Fazli was denied by the goalkeeper before Hamrang's header struck the right upright.
The hosts had their first real chance just after the half-hour mark when Mohamed Surour sprung the offside trap but took a poor first touch allowing Almir Tolja to smother the ball.
But they took the lead five minutes before the break when Ali Latifi brought down Midavoodi in the box and skipper Khamis stepped up to coolly slot home the penalty into the left corner of the net.
Al Ahli then extended their lead in stoppage time, once again from the penalty spot.
This time an Ahmed Khalil free-kick was handled in the area and Al Hawasin slotted home the spot-kick to the left of the goalkeeper.
Al Ahli extended their lead to 3-0 in the 66th minute when Midavoodi broke down the right and chipped an effort over the goalkeeper.
It looked as though that was it for Saba but the Iranians responded immediately to reduce the deficit when Yousefi sent a deep cross for Nori, who headed the ball back across the goal and Fazli was on hand to score.
The Iranians then stunned the hosts with two goals in the space of a minute when first Shafiei scored after being put clean through before Nori equalised.
Yousefi then struck with eight minutes remaining when he swiped the ball home from inside the box before Nori squared a pass for Hamrang to fire beyond the goalkeeper and complete the impressive win.
Al Ahli's misery was complete late on as Saad Surour was shown the red card for a reckless challenge.