Thabo Nemukondeni opened the scoring for the away side in the 12th minute, burying the rebound after Thabiso Nkoana's initial shot came back off the upright after a swift counter-attack.
SuperSport equalised in the 26th minute when Michael Morton produced a composed left-footed finish after Washington Arubi fumbled Bradley Grobler's knock-down from Thato Mokeke's pass.
Tuks took the lead again early in the second half when Ian Chikohwa powerwed a header past Ronwen Williams from Lennox Bacela's right-wing cross.
And Chikohwa made another telling impact just minutes later as his wicked, swerving cross forced ex-Tuks man Grant Kekana to score a diving header past his own goalkeeper, and it was 3-1 before an hour played.
But Stuart Baxter's men kept fighting and pulled a goal back in the 73rd minute, top scorer Jeremy Brockie scoring with a header at the near post from Luvolwethu Mpeta's left-sided cross.
And the comeback was completed in the 84th minute when substitute Morne Nel was in the right place at the far post to bundle in Morton's cross after the hard-working midfielder did well to keep an unsuccessful set-piece alive.
The game went into 30 minutes of extra-time and the game took on a familiar shape, United doing the majority of the pressing but Tuks playing cleverly and counter-attacking at speed.
There were no further goals after the first half of extra time, although Williams had to be alert to tip over a fierce shot from Khethukuthula Zwane.
Right at the start of the second period of extra time Arubi was on hand to make a big save when following good work from Brockie outside the box, the ball was poked through to Nel on the penalty spot, but the goalkeeper stood firm.
There was one more controversial moment when, in stoppage time at the end of 120 minutes, Brockie stuck a leg out to block a throw-out from Arubi, and the ball rebounded over the line. Instead, the striker was booked by referee Daniel Bennett as the match went to a penalty shootout.
Williams was the hero in the shootout for SuperSport, saving two penalties, from Botshelo Mfulwane and Partson Jaure, as Matsatsantsa fought through to the quarter-finals.