Zverev puts aside off-court issues to battle on at Australian Open
The seventh seed was made to grind on Margaret Court Arena before toppling the world number 62 4-6, 6-3, 7-6(3), 6-3.
His appearance came a day after a Berlin court said he would face trial in May for allegedly assaulting his former girlfriend in 2020. He denies the accusation.
The 26-year-old was in October ordered to pay a 450,000 euro (£387,400) fine over the incident, but he contested the ruling and the case will now go to court.
Fresh from helping Germany win the United Cup ahead of the Australian Open, Zverev kept his focus on tennis matters in Melbourne.
"Credit to him because he played fantastic," he said of his countryman.
"In the beginning I obviously wasn't playing my best. I had a great tournament at the United Cup, but it's very different coming here and playing at a Grand Slam.
"But you have to focus on yourself, your own shots and finding your rhythm and I did in the tiebreaker and from then on I played much better."
After losing the first set, he reset to take the second with a single break.
There was little between them in the third set, which went to a tie-break, where Zverev's experience came to the fore, winning it with a backhand volley.
Koepfer kept battling, but his more illustrious countryman was always in charge as he powered home.
