The Nigeria international struck 12 minutes after Efkan Bekiroglu had stunned the home crowd with an early sixth-minute opener, sparking the comeback for Okan Buruk’s side.
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Osimhen’s goal marked his third league strike of the campaign and his second in as many outings, underlining his growing influence for the Super Lig champions.
Galatasaray extended their lead in the second half through goals from Gabriel Sara and Mauro Icardi to secure all three points.
The victory keeps Osimhen’s club at the top of the table with 28 points, five clear of second-placed Trabzonspor, while Göztepe remain in fifth position on 16 points.