The midfielder's goal in the last seconds of first-half stoppage time was enough to record an historic 1-0 win in Warsaw despite a late fightback from their opponents.
The 35-year-old praised his team's spirit as they went through to the last eight alongside group winners Czech Republic.
"We didn't have to prove anything to the people,” Karagounis told reporters after the game.
“We have achieved results in our history that other nations probably wouldn't have. We knew that we would play against a strong opponent but we showed our character and our qualities.
"Our fans were really fantastic. We just wanted to make the Greek people happy today.
“I can compare this qualification for the quarter-final with the glory in Portugal eight years ago. I hope that the Greeks have smiles on their faces tonight!"
Sofoklis Pilavios, the president of the Greek Football Association, also praised his country’s efforts in going some way to relieving their Euro 2004-winning campaign in Portugal.
“We are all happy," he declared to the press. “I apologise for not being able to describe what happened. Many congratulations to all the lads.
“It is a very emotional day for Giorgos. We are among the best teams in Europe and this is something important, especially this time.
“It was something that we deserved.”