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Bafana Bafana was frustrated by Namibia for large periods at Al Salam Stadium but finally found a way through in the 68th minute.
Zungu nodded in a corner to ensure South Africa joins Cote d'Ivoire on three points after two games, although Morocco remains in control of Group D after claiming successive victories.
A low-quality first half was summed up by Petrus Shitembi dragging a poor shot wide of the target after a promising Namibia attack, while at the other end Hlompho Kekana was off the mark with an ambitious long-range effort.
Chances were also few and far between after the interval but Percy Tau should have done better after finding space in Namibia's box, only firing a tame strike straight at Lloyd Kazapua.
Buhle Mkhwanazi cleared Deon Hotto's effort off the line in the 65th minute as Namibia briefly threatened, but it was South Africa that clinched the points shortly after.
A left-wing corner from Tau was misjudged by Kazapua and Zungu was left with the simple task of arriving at the back post to head the ball into the unguarded Namibia net.
Namibia's slim chances of qualification now rest on its Group D closer against Cote d'Ivoire, while South Africa wraps up its group-stage campaign against leader Morocco.
