Jeremy Brockie gave SuperSport the lead in spectacular fashion in the 72nd minute with an unstoppable left-footed volley that arrowed into the bottom-left corner from Thabo Mnyamane's outswingng corner-kick.
But Ramahlwe Mphahlele equalised just five minutes later with a towering header at the far post from substitute William Twala's excellent left-wing cross.
The first half was a dire affair as SuperSport sat tight and tried to hit Chiefs on transitions, with neither side able to conjure up any clear scoring opportunities.
However, the second half was a more lively affair towards the end and it started when Itumeleng Khune made a reflex save to deny Mnyamane one-on-one, although the shot was straight at him when he should have scored.
Khune then denied Brockie from close range in the 71st minute, tipping his header onto the roof of the net. But it was from the resultant corner-kick that the striker – unmarked on the edge of the area – opened the scoring.
Brockie tried another volley in the 88th minute but this time Khune stuck a foot out and made the save from a tight angle as the match ended 1-1 and went into extra time.
Chiefs had a golden opportunity to win the match during extra time when Dean Furman was adjudged to have handled the ball in the area, referee Victor Gomes pointing to the spot. But Bernard Parker's weak spot-kick was saved by Reyaad Pieterse.
After it ended 1-1 in 120 minutes, Pieterse also psyched out George Maluleka in the shootout that followed as he struck the upright. Brockie scored the final penalty as SuperSport won the shootout 5-3 to advance to the semi-finals.