Ibragimov shared first place with Vladimir Kramnik and Andrei Kharlov at the 1991 USSR Under-26 Championship. He was awarded the title of Grandmaster by FIDE in 1993. Ibragimov won the Chigorin Memorial in 1994 and the Biel Masters Open tournament in 1997. In 2002 he started to play for the United States, where he lived in New Haven, Connecticut. He was a co-winner of the 2004 U.S. Open and shared third place in the 2006 U.S. Chess Championship. Ibragimov played for the US team that took the bronze medal in the 2006 Chess Olympiad.