His best tournament victories include first at Leipzig 1975, Dubna 1976, Yerevan 1980, Banja Luka 1981, Sochi 1981 and Minsk 1982. He was co-winner of the 1978 Soviet Championship (with Mikhail Tal) and winner of the 1986 Championship. He beat some world champions: Vasily Smyslov at the Moscow Spartakiad 1974, Tal at Sochi 1970, and a young Garry Kasparov at the 1978 Soviet Championship. Tseshkovsky himself almost qualified for the World Championship candidates matches when he finished fourth in the 1976 Manila Interzonal, one place lower than was needed to progress to the next stage. At the 27th Chess Olympiad in 1986, he scored 2½/5 as the second reserve board to help the USSR team win the gold medal.