Australia's doubles pair of Matthew Ebden (35) and Max Purcell (25) secured their team's progress with a 6-4, 7-5 victory over Jiri Lehecka (22) and Adam Pavlasek (29) in the decider after Alex de Minaur (24) edged past Lehecka 4-6, 7-6, 7-5 to level the tie at 1-1.
"I've never seen anything like it live," Ebden said of their comeback victory. "I've seen it on TV back in the day, comebacks when your backs are against the wall. But what Alex did to give us the chance to come out here and fight was incredible.
"I was lost for words. We were inside thinking we're one game away from being out of here and... winning and going to the semi-finals, we couldn't be more proud.
"I think that really is the sign of our team spirit and our fighting culture. Can't wait for the next one."
The Czechs, who arrived at the Finals with a 9-0 win-loss record from three Group Stage ties in September, were seeking a first victory over Australia since 1975 and Tomas Machac (24) beat Jordan Thompson (29) 6-4, 7-5 to give them a 1-0 lead.
Yet it was not to be their day as Australia, who lost to Canada in the 2022 final, remained on course for a 29th crown in the premier men's team competition to get closer to the United States who are top of the pile with 32 titles.
Up next for Lleyton Hewitt's side in Friday's semi-final are outsiders Finland, who defied the odds to end Canada's reign as champions a day earlier.
Machac, the Czech number two, backed up an early break with a comfortable service game and withstood a strong fightback from Thompson at 5-4 in the first set of the opening match.
There was little to separate the hard-hitting pair in the second set up to 4-4 and Machac traded breaks with his opponent before edging ahead to seal the victory with a forehand winner.
Australia were without Thanasi Kokkinakis (27), who was forced to withdraw from the Finals with an ankle injury.
Later, Australian Open quarter-finalist Lehecka saved five break points in the 10th game of the second singles contest and won a tight opening set with a big serve but surrendered the next amid a late charge from De Minaur in the tiebreak.
Lehecka regrouped in the decider and could have gone 4-2 up but an inspired De Minaur kept chipping away and raised his level towards the end to prevail.
"It would have been quite easy for me to roll over under the circumstances, but that's just not the type of player I am," De Minaur said.
"I'm always going to fight until the end... I got a little sniff. I was able to roll with it. I think my reputation's quite strong when it comes to these types of situations. It precedes me. Maybe I won that match because of that today."
Italy will take on the Netherlands before Novak Djokovic's Serbia meet Britain in Thursday's quarter-final double-header.
