Kolarov turned provider for Dzeko minutes later, but the striker could only poke his effort straight into the arms of the goalkeeper, who had an unusually quiet opening 30 minutes of football.
Sameehg Doutie came close to sending the home side into the lead in the latter stages of the half, but the 24-year-old was denied by a last-ditch challenge from Kompany.
It was Supersport who took the lead early in the second half when Mame Niang fired a powerful header into the back of the net from close range.
Both sides made changes late in the match, with City midfielder James Milner coming close to levelling the scores after 74 minutes, but his powerful effort was cleared off the line.
The former Premier League champions continued to force the issue in the latter stages, but it was the home side who added their second late on courtesy of Kermit Erasmus.