Barty won the opening set against Slovakian 15th seed Dominka Cibulkova at Melbourne Park before being well beaten in the next two sets.
And Ebden was two sets to love up against Russian Mikhail Youzhny before losing in five.
Earlier defending champion and men's top seed Novak Djokovic, Australia's Sam Stosur, Maria Sharapova and Venus Willilams all made it safely through to the second round.
How the Australians fared on day one on Monday (prefix denotes seeding):
Men's singles, round 1:
Matthew Ebden lost to 23-Mikhail Youzhny (RUS) 4-6 6-7 (0-7) 6-2 7-6 (4-7) 6-3
Women's singles, round 1:
9-Samantha Stosur bt Chang Kai-Chen (TPE) 7-6 (3) 6-3
Bojana Bobusic lost to 4-Agnieszka Radwanska (POL) 7-5 6-0
Sacha Jones lost to Kristyna Pliskova (CZE) 6-3 6-2
Olivia Rogowska lost to Vesna Dolonc (SRB) 5-7 7-5 8-6
Ashleigh Barty lost to 15-Dominika Cibulkova (SVK) 3-6 6-0 6-1