Last season's Premier League runners-up have lost two of their opening eight games and currently trail table-toppers Arsenal by six points, although a hard-fought win over CSKA Moscow has boosted their Champions League hopes.
And ahead of Sunday's clash with Chelsea, Zabaleta has outlined the new manager's mission plan for the season.
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"The Premier League and the Champions League are the most important options for every team, but he is also serious about winning the FA Cup and the Capital One Cup. It’s what you should be aiming for when you want to be the best.
"Last season hurt us. We finished second to United and lost the FA Cup final and this club cannot accept silver medals. That experience has made us more focused, I think.
"We have a better squad now than when we won the title in 2012. The manager has a lot of options. The new signings have all brought something different."
City have won six of their last nine meetings with Chelsea, and a win would see them move above their title rivals into third place, but Zabaleta feels the presence of Jose Mourinho means the form book counts for nothing.
"Chelsea is now a huge game for both teams," he added. "We had a good record against them last season, but that will mean nothing this time because Jose Mourinho is back."
David Silva, meanwhile, is certain Pellegrini's tactical approach has benefited the team and results will improve over time.
He said, as quoted by the Guardian: "The manager likes to play an open, attacking style of football, and that suits me very well. There is still room for improvement ... but things are heading in the right direction. It is nice to have an objective and a clear idea of what the manager is trying to achieve.
"Mourinho always sets his team up to be hard to beat. If you play one of his sides you know you will be up against difficult opponents."