The Turkish club host Juventus in their final Group B game in midweek and need a win to progress to the knockout stages, whereas a draw would be enough for the Italian side.
"What worries me is that Juve can be happy with a draw and still qualify," Mancini told Tuttosport.
"We have the utmost respect for them, as they are one of the best teams in Europe, getting better every year and it will be tough for us. I think we need to be calm when playing football, not tense or full of doubt.
"What we must first of all do on Tuesday is step on to that pitch aware that we have a chance and can gamble with our chips. We are playing at home, which is an advantage, and anything can happen in a football match."
Mancini joined Galatasaray in September after spending three-and-a-half years at Manchester City.