The 2008 Olympic champion needed just one hour and 16 minutes to complete a comfortable 6-2 6-4 win over Gajdosova on Tuesday.
Dementieva will now meet Polish sixth seed Agnieszka Radwanska who moved into the last eight with a 6-3 7-5 win over Slovakian Daniela Hantuchova.
The No.1 seed Serena Williams won comfortably 6-1 6-2 over Italian Sara Errani to set-up a quarterfinal meeting with the in-form Caroline Wozniacki.
Russian second seed Dinara Safina has also advanced with an easy 6-3 6-0 victory over qualifier Vera Dushevina.
Serena Williams (USA) w Sara Errani (ITA) 6-1 6-2
Ai Sugiyama (JAP) w Anabel Medina Garrigues (SPA) 2-6 6-1 6-2
Dinara Safina (RUS) w Vera Douchevina (RUS) 6-3 6-0
Elena Dementieva (RUS) w Jarmila Gajdosova (SLO) 6-2 6-4
Svetlana Kuznetsova (RUS) w Kaia Kanepi (EST) 6-2 2-6 7-6(3)
Agnieszka Radwanska (POL) w Daniela Hantuchova (SLO) 6-3 7-5
Caroline Wozniacki (DEN) w Melanie South (ENG) 6-2 6-0
Alize Cornet (FRA) w Alisa Kleybanova (RUS) 7-6(5) 6-1
011309 tuesday/ photo via Medibank International Sydney