Tschesnokow started professional tennis in 1985. He reached his highest world ranking position (9th place) in 1991, in that year he also won the tournament in Montreal , which is part of the Tennis Masters Series . He celebrated his first victory at a Masters tournament in 1990 in Monte Carlo , where he defeated the then clay court king Thomas Muster in the final without losing a set.
Between 1987 and 1990 he won another five titles in tournaments on the ATP Tour . His greatest success at a Grand Slam tournament dates back to 1989 - at the French Open , Chesnokov advanced to the semi-finals, but was then defeated by the eventual champion Michael Chang .