Zverev gave Germany the early advantage with a gruelling four-hour win over Alex de Minaur 7-5, 4-6, 4-6, 6-3, 7-6 (7/4), before Kyrgios disposed of Jan-Lennard Struff in straight sets 6-4, 6-4, 6-4.
The mercurial Kyrgios, who has responded well to the Davis Cup captaincy of Lleyton Hewitt, was in sublime touch against Struff.
The German battled hard but he was a class below the Australian and succumbed in 97 minutes in front of a large and passionate crowd at the Pat Rafter Arena.
"He's a tough opponent to play against, he doesn't really give you any rhythm so I thought I did a good job of getting it done pretty quickly and giving myself the best chance to recover, Kyrgios said.
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Earlier, Zverev outlasted de Minaur in a four-hour marathon.
The world number five had looked down and out when de Minaur, playing superb defensive tennis against the more attacking German, took a two sets to one lead.
But Zverev fought back to win the fourth set before recovering from an early break in the fifth to force a tiebreak.
"I was more aggressive than him from the baseline," Zverev said.
"I think over five sets the one who is more aggressive probably wins. The other one might tire a little bit and get a bit slower."
Zverev, 20, has a disappointing Davis Cup record, with one win and four losses before Friday's match.
Considered one of the rising stars of the men's game, he also has a poor Grand Slam record and fell in the third round of the Australian Open.
De Minaur said Zverev's experience on the big stage helped the German.
"It was all about him being able to get free points on his serve which was decisive," he said.
"He's a big guy, a top 10 player and in the decisive moments he was able to come up with a couple of free points.
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