Dellacqua opened the Fed Cup quarterfinal round with a 6-0, 6-2 win over Irina Khromacheva and Stosur followed with a 6-4, 6-0 decision over Veronika Kudermetova.
A win in either of Sunday's reverse singles or the series-ending doubles match will advance Australia to the Fed Cup semifinals in April against either Germany or the Slovak Republic.
Stosur is to take on Khromacheva in Sunday's first match, with Dellacqua and Veronika Kudermetova meeting in the second. The current doubles schedule has Dellacqua and Ashleigh Barty going against Russians Victoria Kan and Valeria Solovyeva.
Russia fielded a very inexperienced Fed Cup team due to injury and other commitments by top players -- world No. 5-ranked Maria Sharapova, for example, took part in the Winter Olympics opening ceremonies Friday in Sochi, Russia.
Dellacqua need only 58 minutes to dispatch Khromacheva, winning more than 60 percent of the points overall. Khromacheva had nine break opportunities -- all in the second set -- but converted only one. Dellacqua was 6-of-11 in break-point situations.
Stosur -- off the court in an even more brisk 57 minutes -- allowed Kudermetova only two break points and was 24-of-31 points in the second-set shutout.