Barty opened proceedings in the evening at RAC Arena in Perth with a straight sets win over Alize Cornet.
Cornet, who won the mixed-teams tournament with Jo-Wilfried Tsonga in 2014, was under pressure from the 22-year-old world No.15 at regular intervals before Barty claimed the match 7-5 6-3.
Ebden then secured the match for Australia by defeating world No. 32 Lucas Pouille.
The 31-year-old underdog had to come back from a set down and survived a tight second set tie-breaker to win 3-6, 7-6 (5), 6-2.
France won the dead mixed doubles rubber leaving the final score 2-1 to Australia.