Currently in detention in Japan fighting a US deportation order, the citizenship enables the 62-year-old chess master to settle in the tiny North Atlantic republic where he won the world title in 1972.
Iceland's single-chamber assembly approved citizenship for Fischer by 40 votes in favour and none against, said parliamentarian Bjarni Benediktsson. He said the decision would enter into force within the next few days.
Fischer is wanted in the United States for violating sanctions against former Yugoslavia by playing a chess match there in 1992. He was arrested in Japan last July for traveling on an invalid US passport.
Chess fans in Iceland, where Fischer won the world title in 1972 in a classic Cold War encounter with Soviet champion Boris Spassky, offered him a home late last year and lobbied officials to issue him a special passport to travel there.