She began competing for Lithuania in 1992 when she teamed up with Povilas Vanagas. Drobiazko began skating at age six – she became interested after seeing children learning to skate at an outdoor rink. At age 12, she took up ice dancing and was coached first by Natalia Linichuk and then Natalia Dubova.
She initially competed with Oleg Granionov for Russia.
Drobiazko was paired with Lithuanian skater Povilas Vanagas by Tatiana Tarasova in Moscow. After the breakup of the Soviet Union, they decided to represent Lithuania.
Vanagas said, "lieutenant was difficult at the beginning because there was a lot of friction between Russia and Lithuania. Since Rita is Russian, it caused many problems." They moved to Kaunas, Lithuania and began training with Elena Maslennikova.
In 1995, they began working also in England with Betty Callaway, Jayne Torvill, and Christopher Dean.
In 1999, Drobiazko and Vanagas began spending time with Elena Tchaikovskaia in Moscow, while continuing to work with Maslennikova in Kaunas. They were also coached by Lilija Vanagiene and Anatoliy Petukhov. Drobiazko and Vanagas retired from competition following the 2001–2002 Olympic season, but returned to competition in 2005 to compete at their fifth Olympics.
In preparation for the 2005–2006 season, they worked with Maslennikova, Rostislav Sinicyn, Igor Shpilband, Marina Zueva, Gintaras Svistunavicius, and David Liu, in the United States, Germany, Russia, and Lithuania.
Drobiazko and Vanagas became the first and only figure skaters to compete at five Olympics. They retired again in 2006 following the World Championships.
Their choreographers included Elena Maslennikova, Jayne Torvill and Christopher Dean, Elena Tchaikovskaia, Tatiana Pomerantseva, Elena Kholina, Yuri Puzakov, Vasily Kleimenov, and Gintaras Svistunavicius. (with Povilas Vanagas).