Marin Cilic produced a thrilling fightback in the second rubber against home favourite Pablo Carreno Busta to win 5-7 6-3 7-6 (5) and silence a partisan crowd.
After Borna Coric had beaten Roberto Bautista Agut 6-4 7-6 (4) in the opening rubber to edge Croatia 1-0 in front, Cilic hit back from 4-1 down in the deciding tie-break to triumph in three hours and 13 minutes.
The 34-year-old world number 17 had also been two points from defeat at 5-6 down in the third set before staging his comeback.
"It was an incredible battle," said Cilic.
Australia booked their place in the last four on Tuesday when they won 2-0 against the Netherlands.
Both Jordan Thompson and Alex de Minaur recovered from a set down in their singles matches to beat Tallon Griekspoor 4-6 7-5 6-3 and Botic van de Zandschulp 5-7 6-3 6-4 respectively.
"We've got a great team, a lot of belief and a lot of passion," De Minaur said after his victory.
"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."
In the other quarter-final matches, Italy take on the USA and Germany face Canada on Thursday.