Zverev produced some superb tennis to down Kyrgios 6-2, 7-6 (7/3), 6-2 in one hour, 48 minutes and put Germany into a quarter-final against either Spain or Great Britain.
"(To win) against a very strong Australian team makes us very confident for the next round, for the upcoming years," German captain Michael Kohlmann said.
"We showed we have a lot of good players, a lot of strong players.
"We showed that we are able to go further than this."
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He also returned well, getting many of Kyrgios's thunderbolts back in play and then winning the battle from the back of the court.
"It's awesome, it's an amazing feeling and without my teammates it wouldn't have been possible," Zverev said.
"Obviously we're happy but hopefully this is just the beginning for us."
Kyrgios went into the match full of expectation after an impressive win over Jan-Lennard Struff in Friday's opening singles.
Zverev, on the other hand, had looked out of touch on Friday and was pushed for almost four hours before seeing off 18-year-old Alex de Minaur in five sets.
The mercurial Australian opened brilliantly, holding his serve to love much to the delight of the boisterous home crowd.
But that was almost as good as it got for the Australian team, as Kyrgios lost two of his next three service games to surrender the opening set in just 23 minutes.
Kyrgios appeared troubled by an elbow problem in the first set and it became more noticeable in the second, the Australian often shaking his right arm between points.
The Australian continued to serve well and had two set points on Zverev's serve at 6-5, but the German saved both then played a superb tiebreak to take a stranglehold on the rubber.
Once Zverev broke to go ahead 3-1 the result was never really in doubt and at 2-5, Kyrgios was broken again to give Germany the tie.
Kyrgios said he felt a problem with his elbow after Friday's match.
"It obviously affected me a lot," a downcast Kyrgios said.
"It's tough to go out there and not be able to put in your best performance.
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