The reigning English champions were unsuccessful in their pursuit of the knockout phase of Europe's premier club competition last season and Mancini fears they will experience a similar fate this term if they don't sharpen up soon.
Ahead of tomorrow night's home meeting with German champions Borussia Dortmund, Mancini knows City have to get better in their group which also contains La Liga champs Real Madrid and Dutch title-holders Ajax.
"This is the second year for us in the Champions League and the second year that we got the worst group," Mancini said.
"Like we have improved (over the years) in the Premier League, we need time to improve in the Champions League, but we don't have a lot of time - only five more games, and we need to improve quickly.
"We need to not concede easy goals to drop points like we did against Real Madrid (3-2 loss a fortnight ago). If we want to go through, we need to improve very quickly."
He added: "The Champions League is a difficult competition, but football is the same. It doesn't change - it is still 11 players against 11.
"The problem is that you are always playing against the best team from another country and every game is difficult.
"I hope I can improve my record in the Champions League, and I think we have arrived at the time where we can win the Champions League."