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.
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.
Roberto Mancini: "It's difficult to change a lot in five weeks.
"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."
Manchester City Fixtures
Tuesday November 5
19:45 Champions League - Group D
Manchester City v CSKA Moscow
Sunday November 10
14:05 Premier League
Sunderland v Manchester City
Sunday November 24
13:30 Premier League
Manchester City v Tottenham
Wednesday November 27
19:45 Champions League - Group D
Manchester City v Plzen
Sunday December 1
16:10 Premier League
Manchester City v Swansea City
Wednesday December 4
20:00 Premier League
West Brom v Manchester City
Saturday December 7
15:00Premier League
Southampton v Manchester City