Borna Coric gave Croatia the upper hand with a 6-4 6-3 victory over Thanasi Kokkinakis in the first singles rubber, the 26-year-old hitting 19 winners and committing just nine unforced errors.
However, Australian number one Alex De Minaur then beat former US Open champion Marin Cilic 6-2 6-2 in a sensational result - needing just 44 minutes to level the tie and set up a decisive doubles rubber.
"I'm just happy to give our country a chance," said De Minaur. "Very happy with my performance. We play with passion, we play with pride and I know our doubles guys will leave it all out there."
And it was the Australian pair of Max Purcell and Jordan Thompson who held their nerve as they fought back from a set down to beat Nikola Mektic and Mate Pavic 6-7 (3) 7-5 6-4 in two hours and 14 minutes.
While the contest was tight all the way, the Aussies found an extra level from 3-3 in the decider, hitting several decisive groundstrokes, passing shots and volleys to rip through the defence of their opponents.
"There's nothing ***** like it, mate," said Purcell in his on-court interview after the win. "Anything green and gold, there's nothing like it."
Captain Lleyton Hewitt had selected Thompson as a late replacement for Matt Ebden, and the decision paid off handsomely..
"We had a small injury concern for one of our players but I had no concerns about the chemistry of these two [Thompson and Purcell]," Hewitt said.
"It's been a long time. We're a very proud Davis Cup nation and I'm very proud of these guys. They deserve to be in the final and in with a chance of winning this thing."
Italy take on Canada in the second semi-final on Saturday.