"I've been here three years and I've done everything possible, the best I could," the 44-year-old was quoted as saying by the club's website.
Kuban's wealthy owner Oleg Mkrtychan said he was taken by surprise at Petrescu's resignation.
"Right after our win over (city rivals) Krasnodar (last Friday), Petrescu came to me and said he wanted to leave. I had never expected to hear him say this," said Mkrtychan.
"I'm very sad to lose Petrescu but we must respect his decision because he has done great things for this club."
The 2-1 victory in the Krasnodar derby lifted Kuban to eighth place in the table with six points from four games.
Petrescu was named Kuban coach in December 2009 and the following season the side from the Black Sea region won a promotion to Russia's top flight.
He extended his contract until 2015 in November after guiding the club to the top half of the league table.
The new contract included improved terms for Petrescu after he was hailed as "our best player" by the club's bosses.
"I can be proud of all the things we have done here but now I need a new challenge," the Romanian said.
In an interview with Reuters last year, Petrescu said he hoped to return to Chelsea in the future.
"I'll never forget the time I spent at Stamford Bridge so my dream is to return there as coach or manager one day," he said then.