Born in Taldykorgan, Almaty Province, Kivilev began his amateur racing career in Spain, before moving to France, where he wore the EC Saint-Étienne jersey.

[4] Despite his lack of professional victories, Kivilev attracted admirers for his riding style, and despite interest from US Postal Service,[5] signed with Ag2r Prévoyance in 2000, before moving to Cofidis in 2001.

It was at Cofidis that his career started to take off: in his first season, not only did he win the Route du Sud and stage five of the Dauphiné Libéré race, between Romans-sur-Isère and Grenoble,[6] but also had a sensational performance in the Tour de France.

[9] In fact, with later doping scandals eliminating those ahead of him on the podium, the French newspaper Le Monde retroactively (and unofficially) named Kivilev winner of the 2001 Tour de France.

Approximately forty kilometres from the stage finish, as the peloton passed through Saint-Chamond, he collided with his Polish teammate Marek Rutkiewicz and German Volker Ordowski of Team Gerolsteiner.