The Mamelodi Sundowns midfielder came on at the break with the visitors struggling to make a breakthrough against a side who sit 122 places below them in the FIFA World Rankings.
And coach Gordon Igesund's decision to introduce the 21-year-old proved to be an inspired move, as South Africa soon found themselves two up four minutes into the second half.
First, Zungu combined well with fellow substitute Lennox Bacela to send Reneilwe Letsholonyane through on goal and the midfielder made no mistake in slotting the ball beyond Swaziland goalkeeper Nhlanla Gwebu.
And South Africa doubled their advantage a minute later as Zungu rounded Gwebu for a simple finish.
Thabo Nthete added a third after 63 minutes with a low volley as South Africa moved out of sight against their unfancied opponents.
South Africa will next face a step up in class on Tuesday when they welcome world champions Spain to Johannesburg.
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