Kochetkova trained at the Moscow State Academy of Choreography, in the class of Sofia Golovkina for eight years before dancing with The Royal Ballet and English National Ballet in London. She danced with the San Francisco Ballet as a principal dancer from 2007 and joined the American Ballet Theatre as a principal dancer in 2015. Her classical repertoire includes the title role in Giselle, Aurora in The Sleeping Beauty, Kitri in Don Quixote, Juliet in Romeo & Juliet, Odette-Odile in Swan Lake, Clara and the Sugar Plum Fairy in The Nutcracker and the title roles of Alice in Alice in Wonderland and Nikiya in Kingdom of Shades in La Bayadére. She has also performed in George Balanchine's Coppélia (as Swanilda), Divertimento No. 15, Jewels (Emeralds and Rubies), Serenade and Theme and Variations, Symphony in C (2nd movement), William Forsythe's "In the middle, somewhat elevated" and Artifact Suite, Kenneth MacMillan's Winter Dreams, Frederick Ashton's Symphonic Variations, Wayne McGregor's Chroma (opening night for US premiere) and ballets by David Dawson, Derek Deane, Jorma Elo, Mark Morris, Yuri Possokhov, Alexei Ratmansky, Jerome Robbins, Helgi Tomasson, Christopher Wheeldon and Hans Van Manen. Kochetkova has created principal roles in Yuri Possokhov's Diving into the Lilacs, Raymonda Pas de Deux and Classical Symphony, Helgi Tomasson's On a Theme of Paganini and Trio, and Christopher Wheeldon's Within the Golden Hour and Number 9. She performs as a guest artist with the Bolshoi Ballet and Stanislavsky Theaters in Moscow, the Mariinsky and Mikhailovsky Theaters in St. Petersburg, the Tokyo Ballet in Japan including performances as Kitri in Don Quixote for the opening night of the 2009 NBS World Ballet Festival in Tokyo and with the Bolshoi's Reflections project (2011) at the Segerstrom Center for the Arts in Orange County and the Bolshoi Theatre in Moscow. Kochetkova performed the Grand Pas de Deux in San Francisco Ballet's Nutcracker which was broadcast by PBS in 2008 and won the solo gold medal in the NBC series Superstars of Dance which was watched by over 10 million viewers. After appearing as a Guest Artist with American Ballet Theatre from 2013 to 2015, she was named a principal dancer at ABT in June 2015. She performed as a principal with both San Francisco Ballet and ABT beginning 2015.