Zverev was forced to fight every step of the way before seeing off the challenge of the 18-year-old de Minaur 7-5, 4-6, 4-6, 6-3, 7-6 (7/4) on Pat Rafter Arena.
The world number five had looked down and out when de Minaur, playing superb defensive tennis, took a two sets to love lead.
But Zverev fought back to win the fourth set before recovering from an early break in the fifth to force a tiebreak.
Zverev, 20, has had a disappointing Davis Cup record, with one win and four losses before Friday's match.
More From This Section
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.
But he said he taken heart from eventual champion Roger Federer, who spoke to him after his loss in Melbourne and told him to keep working hard and success would come.