The 65-year-old's appointment comes after the big-spending Russian club sacked previous coach Yuri Krasnozhan after just over a month in the job.
Hiddink served as manager of Russia between 2006 and 2010, and under his tenure the national team reached the semi-finals of Euro 2008.
The Dutch coach, who resigned from managing Turkey after their bid to qualify for Euro 2012 ended in a play-off defeat at the hands of Croatia, has been linked with several high-profile appointments, including the England manager's job and a role at Chelsea.
"We discussed Anzhi long enough at the end of last year," Hiddink told the club's website, "so I made the decision quickly. I'm glad I've got the opportunity to work not just on the football team but with friends who have invited me here.
"Especially valuable for me is that the club is building a major plan to develop football in the country. This is what we talked a lot with [club president] Suleiman Kerimov. After conversations with him, I saw at once the major project."
Fabio Capello, fresh from his exit as England boss, had also been strongly linked to the vacant Anzhi job.