Germany, who lead World Cup qualifying Group Four by a point from Russia after seven games, were warming up for their next competitive match against Azerbaijan on Wednesday in Hanover.
Werder Bremen team mates Ozil and Marko Marin, who partnered Gomez in attack, were surprise starters but soon made an impact.
Marin threatened goalkeeper Rowan Fernandez with a low drive after four minutes while Ozil gave a commanding performance in only his third international and created the first goal.
Gomez applied the finish in the 36th minute, tapping in a Michael Ballack cross after some quick passing with Ozil.
The World Cup hosts rarely got out of their half but a minute later Germany keeper Rene Adler was tested when a header by his own defender Arne Friedrich forced a spectacular save.
Adler made several fine saves as he battles for the starting spot with Robert Enke and Tim Wiese, although Enke has already been picked to play in three qualifiers in the next five weeks.
Germany were again on top in the second half but missed several good chances through Philipp Lahm and Lukas Podolski before Ozil added a second goal after they again sliced open South Africa's defence in the 77th with some deft passing.
