The hosts looked threatening in the opening stages of the encounter but were guilty of wasting chances as Bradley Grobler and Jeremy Brockie squandered opportunities in the opening 15 minutes.
The Tunisian outfit were then awarded a penalty when Morgan Gould was adjudged to have handled in the box. Saber Khalifa took the spot-kick and slotted into the bottom corner, 0-1.
Eric Tinkler's charges looked for an immediate equaliser and went close in the 26th minute when Grobler headed wide of the upright.
Grobler had another chance just before the break, but the striker directed his header wide of the upright.
The hosts continued to push for an equaliser in the second half, as Renielwe Letsholonyane had powerful strike saved in the 53rd minute.
SuperSport squandered another chance after 68 minutes, with Brockie failing to hit the target with a header from six yards out.
Mnyamane went close to bringing SuperSport level on 80 minutes, but his effort from distance was saved by the goalkeeper.
SuperSport were awarded a penalty in the final minutes of the contest. Mnyamane stepped up and made no mistake from the spot.
SuperSport: Williams, Nhlapo, Daniels, Gould, Modiba, Furman, Letsholonyane, Mokoena, Phala, Brockie, Grobler.