The Italian head coach has overseen two matches in the competition since being appointed in September as successor to Fatih Terim, who departed to manage Turkey.
Both have been hugely encouraging, with Galatasaray coming from behind to draw away to Juventus in the first and following up with a 3-1 home win over FC Copenhagen.
The results have taken the sting out of an opening 6-1 home defeat to Real Madrid, and leave the Turkish champions second in Group B, ahead not only of FC Copenhagen but also Italian giants Juventus.
Juventus have managed just two points so far and face Real Madrid on Tuesday night, when Galatasaray and FC Copenhagen resume their rivalry in the Danish capital.
Mancini's Manchester City missed out on the last 16 in the last two seasons, so to achieve the feat at his new club would be an achievement to savour.
"It's difficult to change a lot in five weeks," Mancini said. "We need time. We've been working very hard.
"Although it seems that Juventus and Real Madrid are favourites in this group, we will try to get out of it.
"I have high expectations. We want to continue having good results."
Galatasaray could build a five-point lead over Juventus by winning at the Parken stadium, but they will be without injured goalkeeper Fernando Muslera (broken toe) and midfielder Wesley Sneijder (thigh). Both suffered setbacks in the Super Lig match against Konyaspor on Friday.
Their Danish hosts have taken just one point from three games so far, meaning their hopes of reaching the last 16 are slight already.
They were swept aside in Istanbul by a rampant Galatasaray last time, with Mancini's team building a three-goal lead by half-time. FC Copenhagen's goal came as a very late consolation.
Head coach Stale Solbakken said on Monday: "Galatasaray were the better team in the first game but I'm 100% certain tomorrow's will be a more normal and close match with few chances.
"They were on fire. I think most fans and others around the team believe the first half they played against us and the second half they played against Juventus were among the best they have played.
"It was more or less the first 20-25 minutes that killed us there."
Solbakken revealed, in a colourful way, that how he sleeps before the match could influence his team selection.
"There will be a few changes, and it depends on whether I sleep on my stomach or on my back," he said.
"Players could show up in my dream and score a goal, or they could make it a nightmare and smoke on the bench."