In earliest international competition he was a joint winner at Decin 1974, shared first place (with Lev Polugaevsky) at Sochi 1976 and won category 8 tournaments at Le Havre 1977 and Cienfuegos 1979. At Novi Sad in 1979, he shared second prize with Efim Geller behind Florin Gheorghiu. At Wijk aan Zee in 1981, he shared 3rd place and in 1983, was joint champion of Moscow. Sveshnikov won the Latvian Chess Championship in 2003 and 2010. In 2017, Sveshnikov won the 65+ section of the World Senior Chess Championship.