The Bundesliga side could only draw 0-0 at the Mercedes-Benz Arena, a stalemate which left Bruno Labbadia's side bottom of Group E.
However, Stuttgart go into the match unbeaten in five matches, the last game of that run a goalless draw against reigning German champions Borussia Dortmund.
"They will play a bit more (than in the first game)," he told the official Stuttgart website.
"It was the last pass in the first leg which let us down.
"We have to do better because when you play against such a team you can hardly play through the middle and more must come down the flanks.
"Therefore, crosses from the flanks work relatively well. I hope that Copenhagen have a bit more possession, then we may play more counter-attacking football, which is one of our strengths.
"For us it was a bit of a problem to play against such massive defensive formations. We were sometimes not snappy enough (without passes).
"Hopefully we have learned from it. We now have really good form with the draw in Dortmund.
"So we have more confidence, because the foundation is good defensive play. We now have a better plan, everybody knows what they need to do."
Kvist spent seven years with Copenhagen and is excited about returning to the Danish capital.
"You could say that I'm really looking forward to it, for me this is very special, Copenhagen is my city," he said.
"I am also pleased to play again in the Parken Stadium, although I've already done so with the national team, but it's different.
"And of course, I am delighted to be back in the city and to see the fans."
Copenhagen coach Ariel Jacobs is expecting a tough match.
"It continues to be representative of a strong European league, the Bundesliga," he told reporters.
"We must prepare ourselves. I would not say that we should show respect, but we must fight to get the respect we deserve."
Copenhagen could have striker Nicola Jorgensen fit after missing the last month with a hamstring injury.
"It is up to the coach whether he will use me," he told FCK TV. "If I get five or 10 minutes, it's not that important. The most important thing is just to get started again.
"It is clear that it will take a few matches. I just hope it goes well, and there will be no reaction."