Drăgoi began gymnastics at the age of four at CSS Buzău, training under Costică Cliniciu and 1976 Olympic silver medalistGabriela Trușcă. In 2007, she placed fourth in the all-around at the 7th Lugano International Trophy and eighth all-around, sixth on vault, eighth on uneven bars and fourth on floor at the 27th International Junior Competition in Yokohama.
Drăgoi's first major international competition was the 2008 European Championships, where she competed in the team finals after Cerasela Pătraşcu injured her foot during qualifications. Drăgoi helped the Romanian team win the gold medal by competing on vault and uneven bars. At the 2008 Romanian Nationals, Drăgoi placed first on beam, ahead of the more experienced Steliana Nistor. She also placed seventh in the all-around and sixth on vault, and was selected—along with Nistor, Sandra Izbaşa, Andreea Grigore, Andreea Acatrinei, and Anamaria Tămârjan—to be a member of the Romanian team at the 2008 Olympic Games. At the Olympics, Drăgoi helped Romaniaqualify in fourth place to the team finals by competing on uneven bars, beam, and floor. Her score on floor was the lowest among her teammates and was not counted. However, her score on beam was the highest, and she qualified to the beam final in seventh place. In the team final, she helped Romania win the bronze medal by competing on the uneven bars. Individually, she placed fifth in the beam final with a score of 15.625.
2009
Struggling with health problems, Drăgoi was not at her full physical potential at the 2009 European Championships. However, she qualified to the beam finals in sixth place and was ninth on uneven bars. She won bronze in the beam finals behind her teammate Tămârjan and Ukrainian Yana Demyanchuk. In May, she competed at the Glasgow Grand Prix, where she won the bronze on balance beam and placed fifth on uneven bars and sixth on floor. Later that year, she replaced an injured Izbaşa at the 2009 World Artistic Gymnastics Championships. There, she placed 11th and 17th in the uneven bars and beam qualifications, respectively.
2010
In January 2010, Drăgoi underwent surgery for a torn tibia. She made a comeback in August at a friendly meet in Great Britain, where she placed first with the team and fourth on uneven bars. Later that year, she was selected to the Romanian team for the 2010 World Championships. She placed fourth with the team at Worlds.
