Eric Tinkler's charges enjoyed most of the possession in the opening stages of the first half, but found it tough break down the Celtic defence.
Celtic had a chance to take the lead after 25 minutes, but Sipho Jembula somehow headed wide of the upright from six-yards out.
The hosts were made to rue that miss five minutes later when Grant Kekana picked out Aubrey Modiba, who acrobatically fired past Patrick Tignyemb, 0-1.
That proved to be the final chance of the half despite SuperSport's dominance, as the visitors went into the break leading by one goal.
SuperSport extended their advantage 10 minutes into the second half as Jeremy Brockie latched onto Dove Wome's superb pass before volleying past Tignyemb, 0-2.
Goalkeeper Reyaad Pieterse came to SuperSport's rescue after 65 minutes as he produced a fine save to deny Victor Letsoalo from close range.
Celtic were awarded a lifeline in the 77th minute when Teboho Mokoena handled in the box and the referee pointed to the spot. Mbhazima Rikhotso stepped up and slotted home the penalty.
The hosts eventually leveled matters in the 81st minutes as Rikhotso raced to the byline before pulling the ball back for Jacky Motshegwa, who slotted into the bottom corner.
Neither side could find a winner the final stages of the half, which forced the clash into extra time.
However, the final 30 minutes did not prove fruitful and the match went to a shoot-out, where Celtic triumphed 5-4.