De Minaur beat Botic Van De Zandschulp 5-7, 6-3, 6-4 to clinch the series 2-0 and set up a semifinal tie against either Spain or Croatia.
"We've got a great team, a lot of belief and a lot of passion,'' De Minaur said. "I'm very happy to win today. ''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."
Australia is the second most successful nation in the team competition with 28 titles but the last triumph dates to 2003.
De Minaur completed his rally with a break in the 10th game of the final set by building a 40-0 lead and converting the first match point on an indoor hardcourt in the southern Spanish city of Malaga.
He started his comeback by breaking Van de Zandschulp in the first and last games of the second set. The Dutch player took the first set with a break in the 12th game. He recorded three victories in the group stage, including a win over American Taylor Fritz.
Earlier, Jordan Thompson also came from a set down to defeat Tallon Griekspoor 4-6, 7-5, 6-3 and give Australia a 1-0 lead.
The Australian hit 16 aces to overcome his Dutch opponent in a hard-fought match that took almost three hours. Thompson got the decisive break to jump to a 4-2 lead in the final set and held on. After losing the opening set, the Australian came back with a break in the 12th game of the second set.
The quarterfinal tie between the 2019 champion Spain and 2021 runner-up Croatia is Wednesday.
The quarterfinals wrap up Thursday, when the U.S. meets Italy, and Canada plays Germany. The semifinals are Friday and Saturday, and the title will be decided Sunday.