The result lifted the defending champions to the top of the table with seven points from three matches as they continue their bid to retain the title.
Göztepe shocked the home crowd by taking the lead in the 16th minute. Novatus Miroshi finished smartly after a well-worked move to give the visitors a surprise advantage.
Galatasaray responded before half-time. In the 33rd minute, Davinson Sánchez levelled the scores, converting from a set-piece situation to restore parity and settle the nerves of the Rams Park faithful.
The second half belonged largely to the hosts. Gabriel Sara put Galatasaray ahead in the 55th minute, finishing coolly after receiving a clever back-heeled assist from Osimhen. The Nigerian striker then took centre stage himself. After being fouled in the box, he stepped up and converted the resulting penalty in the 63rd minute to make it 3-1.
Göztepe refused to fold. Just two minutes later, Juan pulled a goal back to set up a tense finale at 3-2. Galatasaray had another effort from Osimhen ruled out for offside shortly afterwards, but they managed the remaining time professionally to secure the three points.
Osimhen’s contribution took his tally to five goals in the opening three league matches of the season, underlining his importance to Okan Buruk’s side. The Nigerian international continues to lead the Galatasaray attack with power, movement and clinical finishing.
The victory maintains Galatasaray’s strong start to the campaign and keeps them firmly in the early title conversation. Göztepe, meanwhile, remain near the foot of the table with a solitary point from their first three fixtures.
Galatasaray will next travel to face Başakşehir, while Göztepe host Gaziantep FK. For Osimhen and the champions, the hard-fought win provided another positive step in a season already showing plenty of promise.