Saints stunned the former UEFA Cup winners when they drew 1-1 in Istanbul, but manager Callum Davidson has instilled a confidence in his side which has been backed up by their double cup success.
Now midfielder Davidson is determined to finish off the job in the second leg in Perth on Thursday as Saints bid to reach the Europa League play-offs, where Danish side Randers await the winners.
"I think a performance like that against such a big team and a very good team, you are always going to take confidence," Davidson said.
"But without being arrogant, we got results against Celtic and Rangers last year, you don't win two cups if you are a bad team, you don't finish top six if you are a bad team.
"So we knew, without being over-confident, we were capable of getting a result.
"Not one person in the squad said we were going to beat Galatasaray, we are going to do this, it was just a case of do what we do.
"It was a very physical, tough game in that heat. The first 30 minutes were a totally different tempo, it was hard to get your breath because it was so humid. But once we got past that first half hour we posed more of a threat going forward.
"We are confident without being over-confident. We can't wait to get out there and hopefully see McDiarmid Park rocking and hopefully we can create another wee bit of history.
"We've given ourselves a chance but you want to progress."
The second leg has come at the ideal time given limits on crowds were lifted on Monday, and McDiarmid Park is heading for a sell-out.
Davidson doubts Galatasaray will be fazed by 10,000 Saints fans but he will take inspiration from their return.
"There is probably an extra buzz because Galatasaray are such a big name," the 33-year-old said. "I stay in Perth and there's a buzz about the place.
"We have gone from one extreme to another, from playing in front of nobody to playing in front of as close to a full house as we can get.
"Being honest, it will take a lot to intimidate the Galatasaray players from what I saw last week. Last week in Turkey there were only 9,000 fans there but it's still probably one of the best atmospheres I have played in.
"But I have seen McDiarmid Park before when we get a good gate through and it is a good atmosphere.
"We probably did take fans for granted because it was so normal. For the first month or six weeks last season it was great because we were just buzzing to be playing football again.
"But as time goes on, we had so many big games and it's days like that where you think the fans here would be brilliant. Fans are a massive part of football and it does show us now how important fans are for the game. So we are buzzing they are back in."