The Australian fifth seeds, who lost three grand slam finals last year, were beaten 6-1 6-3 by Hungary's Timea Babos and Croatian Petra Martic.
Babos and Martic upset Dellacqua and Barty, runners-up at Melbourne Park in 2013, thanks to 27 winners.
While Barty exited early in the singles, Dellacqua could be excused for having an eye to her fourth-round match against Canadian teenager Eugenie Bouchard.
Barty, 17, was unable to hold serve once during the 59-minute encounter as Babos and Martic eased through.
They will face another Australian in Jarmila Gajdosova – who teams up with Croatian Ajla Tomljanovic – in the third round.