The Turkish side's dismal UEFA Champions League Group D campaign - that resulted in just one point - came to an end at the Turk Telekom Arena as Lukas Podolski and Aaron Ramsey ran rampant in Istanbul.
Galatasaray - bottom of the group - found themselves three goals adrift inside the first half-hour, with Podolski's third-minute opener and Ramsey's brace virtually ending the match as a contest.
Dutch star Wesley Sneijder netted a consolation goal for Galatasaray two minutes from the end, but Podolski restored the visitors' three-goal buffer in the second minute of injury time.
Tuesday's defeat left Galatasaray winless through the group stage but Hamzaoglu was happy with how his team responded in the second half.
"We conceded an early goal which upset our balance and led to us not playing well," Hamzaoglu said afterwards.
"In the second half we played better and became the dominant team. We struggled to find the goal we needed, however.
"Although we lost, the display in the second half makes us hopeful. I am pleased with my players and thank the fans who came to support us."
Hamzaoglu added: "We want to improve as a group, work harder and aim to become stronger.
"We would like to be successful in the league and the cup. We don't want to let the fans down."
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