The 62-year-old Dutchman has coached Russia since 2006 and Chelsea thanked the Russian Football Union for its "understanding and cooperation."
Hiddink's salary, believed to be around $4.5 million a year, is paid by Chelsea's billionaire Russian owner, Roman Abramovich, through his funding of the National Football Academy. He will continue to fulfill his Russia responsibilities while working at Chelsea.
"Abramovich does so much for Russian football that I wanted to do something back," Hiddink was quoted as saying on the Chelsea website.
"There are two scenarios: it goes bad and they say 'go back east' or it goes well, but I'm not thinking that far ahead," he added.
Hiddink is due to meet the players, most of whom are away on international duty, later in the week.
Before the appointment was confirmed, Hiddink vowed that he would not quit on the Russian national team.
"This would only be for the next two or three months, until the end of the season. I will not leave my job with the Russian national team. It's out of the question. When I took the Russia job it was a long-term project and I don't like to leave it unfinished," he was quoted as saying by the BBC.
The Russian national side has just two World Cup qualifiers before the end of the European season - at home to Azerbaijan on March 28 and away to Lichtenstein on April 1, both of which should be easy wins.
Hiddink became the manager of the Russian national team in 2006 and immediately set about rebuilding a demoralized side that had not managed to get out of the group stages at a major tournament since the break up of the Soviet Union. The Dutch coach brought in a host of new, young players, and Russia were transformed into an attacking, adventurous side, subsequently making the semi-finals of Euro 2008.
Despite some initial resistance in Russia to having a foreigner in charge of the national team, the vast majority of the country has now been won over to Hiddink. Indeed, such was the affection felt for the Dutchman that babies were named in his honor after the side's 3-1 victory over the Netherlands in the quarterfinals of Euro 2008.
Hiddink's contract runs through the 2010 World Cup in South Africa, although he earlier said that had not ruled out extending it "if he felt needed."