Born in Tbilisi, Georgian SSR into a family of Georgian Jews, his younger brother is Nodar Djin. Dzindzichashvili won the Junior Championship of the Soviet Union in 1962 and the University Championships in 1966 and 1968. In 1970, he was awarded the title of International Master by FIDE. He left the U.S.S.R. in 1976 for Israel and earned the Grandmaster (GM) title in 1977. One of his best career performances was first place at the 53rd Hastings Chess Festival in 1977/1978, scoring 10½ out of 14 points, a full point ahead of former world champion Tigran Petrosian. In 1979, he settled in the United States, and he won the Lone Pine tournament the next year. He led the U.S.team at the Chess Olympiad in 1984.

He won the U.S. Chess Championship twice, in 1983 and again in 1989, sharing the title with two other players each time. He briefly took up residence in Washington Square Park in New York City, and hustled chess during the 1980s, making a living playing blitz for stakes, as is popular there. He had a cameo in the 1993 film Searching for Bobby Fischer. He also had a brief appearance in Men Who Would Be Kings, a documentary about chess in Washington Square Park set in the 1980s.

Dzindzichashvili is a well-known theoretician and a chess coach. Among his students were 5-time U.S. Champion Gata Kamsky and Eugene Perelshteyn. He is the author and star of multiple chess instructional DVDs entitled "Roman's Lab". He currently produces instructional videos for Chess.com. Topics include openings, middlegames, endgames, famous players, and interesting games.

He is one of the founders of Chess.net internet chess server project, started in 1993. He played third board for the "GGGg" team that won the Amateur Team East tournament in February 2008.^{[citation needed]}