City finished as runners-up in the Premier League and also suffered a shock defeat at the hands of Wigan Athletic in the FA Cup final as they finished the campaign without any silverware.
They have made an inconsistent start to the new season and were beaten 2-1 by Chelsea in the Premier League at the weekend, but Zabaleta says there is a determination in the squad to enjoy great success ahead of Wednesday's Capital One Cup tie at Newcastle United.
"Last season hurt us. We finished second to (Manchester) United and lost the FA Cup final and this club cannot accept silver medals," said Zabaleta.
"The manager always tells us that we have the squad to challenge in everything and that he will always try to put a strong team to win every game.
"The experience against Wigan has made us more focused. This time the manager has told us that we must aim to win all four competitions that we are in.
"I think that's what you want to be aiming for when you want to be the best team."
Pellegrini is expected to make further changes for what will be the club's fourth away game in the space of 12 days, with Joe Hart and Matija Nastasic among those expected to make way following their costly defensive mix-up against Chelsea.
August deadline-day signing Martin Demichelis - the defender who made a belated debut at Chelsea after overcoming a knee injury - could be given another run-out.
Zabaleta, who was the club's player of the season in 2012-13, has been impressed by all the summer arrivals.
He said: "The new signings have all brought something different.
"Alvaro Negredo, Jesus Navas and Fernandinho have settled in very quickly and have already had a big impact.
"Jovetic and Demichelis haven't played a lot of games yet but they will get their chance.
"I think they have made the team much stronger."