Two aces and a winner gave Kyrgios, Australia's No. 1 player, the win over Struff in 1 hour, 37 minutes.
The 18-year-old de Minaur, ranked 139th, was making his Davis Cup debut. He nearly caused a huge upset when he took a two-sets-to-one lead, clinching the third set with an ace.
But the fifth-ranked Zverev came back to take the fourth set, then dominated the tiebreaker to clinch the match in just under four hours.
De Minaur had been Australia's form player of the past month, making the Sydney International final and the Brisbane International semifinals in early January. In both tournaments, he was cheered on by Davis Cup captain Lleyton Hewitt.
"Credit to him. He's playing unbelievable tennis for an 18-year-old, and if he keeps on improving, he'll be great," Zverev said of de Minaur.
The winning country advances to the quarterfinals from April 6-8 against either Britain or Spain. The loser will contest September's World Cup playoffs.
Australia, looking to end a 15-year Davis Cup title drought, has reached the World Group semifinals twice in the past three years.
In other World Group first-round matches:
• Netherlands 1, France 1: In Albertville, France, Richard Gasquet leveled the score for defending champion France by defeating Robin Haase of the Netherlands 6-4, 7-6 (3), 3-6, 7-5 in the first round of the Davis Cup World Group on Friday.
"It's also tough to play when you're 1-0 down. It was a fast surface and he serves well,'' Gasquet said. "He was playing better than me in the fourth set and I had to hold on.''
Mannarino was a late replacement for Jo-Wilfried Tsonga, who withdrew with a knee injury shortly before the draw on Thursday.
Lucas Pouille was due to play the opening rubber but he pulled out on Friday morning with a neck problem.
Italy is bidding to reach the quarterfinals for the fifth time in six years while Japan is looking to end a three-year drought that includes losses to Canada, Britain and France.
The Associated Press contributed to this report.