World No 5 beat the top-ranked Australian in straight sets to set up a last-eight showdown with either Spain or Great Britain.
Zverev in fine form to defeat Kyrgios as Germany reach Davis Cup quarter-finals
German No 1 Alexander Zverev beat Nick Kyrgios in straight sets to give his country a 3-1 win over Australia in their Davis Cup World Group tie in Brisbane on Sunday.
Zverev produced some superb tennis to down Kyrgios 6-2, 7-6, 6-2 in one hour, 48 minutes to 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."
Zverev served flawlessly throughout, only facing two break points in the match, both at the end of the second set and both of which he saved.
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, 20, 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.
But following Germany's win in the doubles on Saturday to give them a 2-1 lead, all the pressure was on Kyrgios, who had to win to keep the tie alive.
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.
He served better in the second set, firing down eight aces, but at 4-3 he had a medical timeout for treatment on his arm.
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," Kyrgios said. "My serve is my biggest strength -- I mean I thought he played great today -- but my serve was not really there and that affects the rest of my game.
"It's tough to go out there and not be able to put in your best performance."