The Citizens have crashed out of European football's top competition in the group stages in their past two campaigns under former boss Roberto Mancini.
Pellegrini said in a press conference: "With Manchester City, we can do better than I did in Spain — although it's not easy to get in the semi or the quarter-final twice.
"I think we can win it. This club needs to do that — at least get passed the first stage and our squad is better than that.
"I know the style we need to play in Europe. It must be different because, in the last two years, the club did not make it out of the group stages."
City will go straight into the group stages of the Champions League after finishing in second place in the Premier League last season.