De Minaur pushed Zverev for nearly four hours in Friday's first singles match but Australian captain Lleyton Hewitt said he had no intention of replacing him with Brisbane's John Millman, who always plays well on his home court.
The World No.14 looked frustrated from the outset on Sunday and required a second set medical time-out for his right arm before succumbing 6-2 7-6 (7-3) 6-2 at Pat Rafter Arena. "It's awesome, it's an fantastic feeling", Zverev said on court after steering Germany to the quarter-finals for the first time in three years.
Australian No 1 Kyrgios was in cruise control for much of the match, in an ominous sign for Zverev before they meet in Sunday's key reverse singles rubber.
"We are in a good position, 2-1 up, but we know it's not all over", Kohlmann said.
"Hopefully this is just the beginning for us", Zverev said after beating Kyrgios.
"It's tough not to put in your best performance".
The victor of the Brisbane tie advances to the Davis Cup quarter-finals in April, either hosting Great Britain or travelling to face Spain.
Holding a 2-0 lead after Friday's singles, Americans Ryan Harrison and Steve Johnson came from a set down to beat Nikola Milojevic and Miljan Zekic 6-7 (3), 6-2, 7-5, 6-4 in front of a partisan Serbian crowd.
"My biggest weapon (serve) was not really there and that affects the rest of my game", he said.
"If you go into a match knowing you had to win otherwise your team loses it is something for sure you are thinking about - 100 per cent".
Germany team captain Michael Kohlmann revealed he was convinced Kyrgios would play doubles and admitted to breathing easier when the Australian No.1 was officially omitted on Saturday.
And the stage was set for what Hewitt had predicted would be an epic clash between tennis' two rising stars Kyrgios and Zverev but it quickly became one-way traffic.
Australian team captain Lleyton Hewitt hinted he might have played in Saturday's doubles but resisted the temptation, pairing Ebden with world number four doubles specialist Peers.
Belgium, finalists previous year, slipped up against Hungary after winning both of Friday's singles matches. The Swiss have now fallen in the first round every year since their maiden title triumph in 2014, which was inspired by 20-time major victor Federer and three-time Grand Slam champion Wawrinka.
Since then Australia have had a 5-6 win-loss doubles record but have lost four of their last seven doubles rubbers. It was doubles partner Tim Puetz who proved the difference, however, in the 6-4, 6-7 (7/1), 6-2 6-7 (7/4), 6-4 win.