Roberto Mancini's side had the best defensive record in the Premier League over the last two seasons but faltered in the Champions League, conceding 11 goals in the group stage as they crashed out.
In contrast, Pellegrini's Malaga only conceded five goals in the Champions League group stages last season, and Kompany is hoping the Chilean's changes at the Etihad Stadium will help them become even tighter.
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"We’ve had a lot of good things in our team. Our organisation and discipline has been really good.
"[Pellegrini] has got his own vision and I think like every big manager he’s got a strong vision about how the team needs to play.
"I think what he’s trying to do is add on to what we have already and just make sure that those little points that we could improve we do improve, maybe recovering the ball quicker and being more dynamic."
Kompany is also confident the arrivals of Fernandinho, Jesus Navas, Alvaro Negredo and Stevan Jovetic will help the team put last season’s disappointing campaign behind them despite the added pressure.
He added: "The first time that we had a [record signing] was Robinho five years ago and the pressure has only been increasing.
"We’ve had a lot of big players. We know how to deal with it and I think our club looks more stable now.
"We’ve always improved. We’ve sometimes had a good season, sometimes a little bad season but you can’t forget where we’ve come from."
Having finished 11 points behind champions United last season, Kompany is relishing the start of the upcoming campaign on August 19 against Newcastle United.
"When you finish second and everyone’s disappointed. That’s a good thing because a few years ago people might have been happy with that," he added.
"I think this squad is just as strong, if not stronger, than the squad that gave us the league title."