Pellegrini took over from Roberto Mancini at the Etihad Stadium after the Italian failed to deliver silverware to the club last season amid allegations of unrest behind the scenes.
Kompany believes that Pellegrini has approached his new position in the right way and hails the fitness levels of the squad despite pre-season having only just started.
"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," Kompany 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 and improve on the last campaign and make sure that whatever the new manager brings in, that we try and make it our own as quick as possible."
City will embark on pre-season trips to South Africa, Hong Kong, Germany and Finland before their opening Premier League fixture against Newcastle United at home on August 17.