He won the Zeeland Open in Vlissingen in 1995. Barsov's victories in Oxford (1998) and York (1999, together with Tiger Hillarp Persson and Julian Hodgson) rate among his most prominent successes in international chess tournaments. In 2001 he won the Hastings International Chess Congress together with Harikrishna and Sasikiran. Barsov competed in the FIDE World Chess Championship 2004, losing in the first round to Alexander Beliavsky. He won a tournament in Saint-Quentin 2004, and Casablanca 2005. Barsov won the Uzbekistani Chess Championship in 2006 and 2010. In 2011, he came first in the 4th Beirut Open tournament. In the following year, Barsov tied for first place with Semetey Tologontegin at the 2nd Central Asia Chess Cup in Bishkek winning the tournament on tiebreak score.