Rubuta ci gaba don wannan sakin layi - Mafi kyawun nasarar gasar da ya samu sun hada da farko a Leipzig 1975, Dubna 1976, Yerevan 1980, Banja Luka 1981, Sochi 1981 da Minsk 1982. Ya kasance mai cin nasara a gasar zakarun Soviet ta 1978 (tare da Mikhail Tal) kuma ya lashe gasar zakarun 1986. Ya doke wasu zakarun duniya: Vasily Smyslov a Moscow Spartakiad 1974, Tal a Sochi 1970, da kuma saurayi Garry Kasparov a gasar zakarun Soviet ta 1978. Tseshkovsky da kansa kusan ya cancanci shiga wasannin 'yan takarar gasar zakarun duniya lokacin da ya gama na huɗu a cikin 1976 Manila Interzonal, wuri ɗaya ƙasa da abin da ake buƙata don ci gaba zuwa mataki na gaba. A gasar Olympics ta Chess ta 27
a shekarar 1986, ya ci 21⁄2/5 a matsayin kwamitin ajiya na biyu don taimakawa kungiyar USSR ta lashe lambar zinare.