She has two siblings, a sister Ágnes, who also was a professional tennis player, and a brother Levente.

[3] Szávay made her professional debut in 2009, at the WTA Tour event in Budapest.

She played qualifications for the singles draw, but lost in the second round to Timea Bacsinszky 6–0, 6–1.

[4] Partnering with her sister Ágnes, she played in the doubles main draw of the Budapest Grand Prix, but they lost to Mariya Koryttseva and Raluca Olaru, 1–6, 3–6 in the first round.

In 2010, Szávay won her first professional title, in doubles, at Durban, South Africa.