The Belgium and City captain reported back to training on Monday and was impressed with how the 59-year-old handled the first session ahead of his debut Premier League season
The former Malaga boss has taken over at the Etihad Stadium this summer, following the sacking of Roberto Mancini in May.
And Kompany feels that Pellegrini has approached his new position in the correct way, while also praising the squad’s fitness on the first day of pre-season training at Carrington.
"I think all of us have done some preparation work during the holidays because the guys look very sharp after a long break, so that's great," the 27-year-old told the club's official website.
"It was also nice to start work with the new manager and I think his approach with the team today was really good.
"It's important now to try to improve on the last campaign and make sure that whatever the new manager brings in, that we try to make it our own as quick as possible."
His side will embark on pre-season trips to South Africa, Hong Kong, Germany and Finland before their opening Premier League fixture at home to Newcastle United on August 17.
City have already recruited Brazilian midfielder Fernandinho from Shakhtar Donetsk and Jesus Navas from Sevilla this summer.