The 65-year-old Dutchman, who has managed the likes of Chelsea and Real Madrid, has signed an 18-month contract with the club, following the sacking of Yuri Krasnozhan earlier this week after just over a month in the job.
"We discussed Anzhi for quite some time at the end of last year so now I took the decision quickly," Hiddink said in a statement on the club website.
"I'm happy that I got this chance - to work not only in a football club but to work with friends who invited me here."
Anzhi are bankrolled by billionaire Suleiman Kerimov and have signed big-name players such as Cameroon striker Samuel Eto'o and Brazilian full back Roberto Carlos.
"I know of the ambitions of the shareholder, the club and the fans - the whole world knows about this now and I will try to do everything so that the team's performances correspond to those ambitions," the new manager said.
Hiddink, among a number of high-profile managers linked with the vacancy including former England boss Fabio Capello, said it was Anzhi's plans to develop the sport in the region of Dagestan where the club is based that had convinced him it was the right move to make.
"This is no less important than the sporting results of the main team and I plan to devote particular attention in this direction," he said.
"Suleiman Kerimov and I spoke about this. From the conversation I had with him I was once again convinced of the scale of the project they were inviting me into."
Russian media reported that Krasnozhan's departure was a result of his inability to deal with Anzhi's top players like Eto'o - a task that someone with Hiddink's track record is unlikely to find so difficult.
As a coach, he won the Dutch championship with PSV Eindhoven six times and in 1988 won a treble of European Cup, Dutch league and cup.
He led his native Netherlands to the Euro 96 quarter-finals and the 1998 World Cup semi-finals. As South Korea coach, he took them to the 2002 World Cup semi-finals, and he later led Australia to the 2006 World Cup finals.
He took charge of the Russia national team and helped them qualify for Euro 2008 at the expense of England and reach the semi-finals of the tournament where they lost 3-0 to eventual winners Spain.
He spent four months with Premier League side Chelsea in 2009 after the sacking of Luiz Felipe Scolari, steering them to FA Cup success.
His last job was as Turkey coach, a post he left in November after their failure to qualify for Euro 2012.
Hiddink's name had been flying around the British media in conjunction with a possible return to Chelsea, who are enduring a poor season by recent standards as the pressure on manager Andre Villas-Boas mounts.
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