Both Jordan Thompson and then Alex de Minaur fell a set down before fighting back to win.
Thompson beat Tallon Griekspoor 4-6, 7-5, 6-3 in 2 hours 24 minutes. The Australian saved break points in three service games in the second set before breaking at 5-5.
De Minaur then beat Botic van de Zandschulp 5-7, 6-3, 6-4 in 2 hrs 40 minutes.
De Minaur's decisive service break at 4-4 in the third gave Australia victory and saved the doubles pairs coming out for a late-night decider.
"We've got a great team, a lot of belief and a lot of passion," said De Minaur. "Every single one of these players on our team is going to play their heart out. We'll do everything we can for our country."
Captain Lleyton Hewitt's Australia will next face either host Spain or defending finalist Croatia who meet on Wednesday afternoon.
